NOTES - ON -THE -SCIENCE- AND 
ART " OF - EDUCATION»NOETLlNG 



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XJNITED STATES OF AMERICA. 



NOTES 



ON THE 



SCIENCE AND ART 



EDUCATION. 



3/ 1;1PR 15 1805 

WILLIAM NOETLIN(^2i:r52^!i^,, nn^ 



PROFESSOR OF PEDAGOGY, STATE NORMAL SCHOOL, BLOOMSBURG, PA. 




NEW YORK AND CHICAGO: 

E. L. KELLOGG &-C0 



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L5ISSS- 



Copyright, 1895, by 

E. L. KELLOGG & CO., 

NEW YORK. 



PREFACE. 



I HAVE for some years dictated my instructions on the 
science and art of teaching, in the form of notes, sugges- 
tions, and hints, to the junior class of this school ; but a 
desire has been expressed by students and others to have 
the work in a more convenient shape for preservation and 
use, and to enable the many teachers who do not have the 
advantages of normal instruction to avail themselves of its 
helps. In compliance with this desire I have prepared the 
notes for publication. They do not, however, constitute 
a methodical or a complete treatise upon pedagogics, but 
only thoughts and suggestions for prospective teachers and 
for beginners in school-room work. 

Every subject has been treated with as much fullness, 
as well as brevity, as my experience has shown necessary. 
To subjects in which beginners need most help I have 
given more space than to others ; this accounts for the dis- 
proportion in the number of pages devoted to different 
subjects. 

The matter will be found in harmony with the best in 
education and teaching, and presented, it is hoped, with a 
sufficiency of explanation to make it intelligible to be- 
ginners. 

Repetitions occur here and there, wherever it is believed 
they will be serviceable to those for whom the book is in- 
tended. 

Wm. Noetling. 

State Normal School, 

Bloomsburg, Pa., 

Dec. 19, 1894, 

3 



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CONTENTS. 



PAGE 

Author's Preface 3 

Introductory Considerations 7 



PART I. 
The Care of the Body 8 

PART II. 

The Mind 11 

Chapter I. The Intellect 11 

Chapter II. The Feelings 33 

Chapter III. The Will 38 

PART III. 
Important Observations and Inferences 42 

PART IV. 

Object Lessons 45 

(a) Their Design 

(d) The Plan or Method of a Lesson 

PART V. 
Penmanship 46 

PART VI. 

Primary Reading 48 

5 



6 Contents. 

PART VII. 

PAGE 

Advanced Reading , 57 

PART VIII. 

Notes and Suggestions on Teaching the English Language. 

Chapter I. General Considerations 64 

Chapter II. i. Oral Language 68 

2. Written Language 71 

PART IX. 

SUGGESl IONS FOR TEACHING NUMBERS 87 

PART X. 
Geography 173 

PART XL 
History 185 

PART XII. 
The Human Body 189 

PART XIII. 
Civil Government 193 

PART XIV. 
Drawing 194 



NOTES ON THE SCIENCE AND 
ART OF EDUCATION. 



Introductory Considerations. 

1. The science of education embraces the principles, 
laws, or knowledge in accordance with which the education 
of a child must be carried on ; and the art, the carrying on 
of the process. In other words, the science is the knowing, 
and the art the doing. 

2. To conduct the education of a child intelligently and 
successfully, the teacher must possess a thorough knowl- 
edge of its constitution, physical and mental. 

3. Understanding the physical constitution implies a 
knowledge of its mechanism, the function each organ per- 
forms, and the relation the various organs bear to one 
another. 

4. Enjoyment is one of the most important elements in 
our existence upon earth ; and this depends upon health — 
the health of the body and of the mind. Preserving health, 
or securing it if not possessed, is therefore one of the first 
things, if not altogether the first, in the education of a child. 

7 



part 91, 

The Care of the Body. 

5. To preserve the body in a healthy state, it must have 
a proper supply, as fast as needed, of the elements that 
enter into its composition ; it must be kept clean, properly 
clothed, must have rest and exercise ; and be surrounded 
by suitable light and proper conditions of atmosphere. 

6. The body is composed of a number of kinds of mate- 
rial, and the quantity of each necessary at any time to pre- 
serve the health and strength of every organ and power 
depends upon the kind of activity in which the person is 
engaged, or, in other words, upon the amount of activity of 
each organ and power. For example, if the muscles chiefly 
are exercised, the material of which they are formed must 
be supplied in sufficient quantity to make up for their 
waste — the amount destroyed by use ; if the bones, the 
material of which they are made must be supplied in suffi- 
cient quantity to make up for their loss by use ; and if the 
brain and nerves, the kind of material that builds them up 
must predominate in the nourishment taken. 

7. The proper nourishment of the body does not, how- 
ever, alone depend upon the quantity of any kind of food 
taken, but more, perhaps, upon the quality and upon the 
preparation it has received. 

8. Another important condition of food is variety. An 
unvaried diet destroys both appetite and health. Walker, 
in his Physiology, says : " The system craves a varied diet, 
and living for a length of time on even an abundance of 
food, if it be unvaried from day to day, will generally result 



The Care of the Body. 



in loss of appetite and in disease. * ^c * When a variety 
of articles cannot be obtained, varied methods of preparing 
and cooking the limited supply should be resorted to. 
* Good cookery means economy; bad cookery means waste.' 
" On the other hand, however, there may be such a thing 
as too great a variety, and this also will destroy the appe- 
tite." 

" It is a dictum of mental as well as physical hygiene that it 
is far better to stint one's self along any other line rather than 
deprive ourselves of food of needed quality and quantity. I 
say stint, for it is not economy. Poor food means poor blood 
and not enough of it, and this in turn means a brain starving 
for oxygen. Such a brain is always a weary brain, slow to re- 
spond and erratic in its activities; and this fatigued, poisoned 
brain can never sustain mental processes of high character or 
strict integrity. Therefore, I say, that in treating of the re- 
ciprocal influence that obtains between body and mind, the 
question of diet is one of special importance to those having 
the care of children, and should be discussed at great length 
by educators in order that it may receive in every quarter the 
attention it so richly deserves." — Krohn, Practical Lessons in 
Psychology. 

9. Exercise, too, at proper times, and suitable in kind 
and amount, is a necessity to health and comfort. But 
exercise of any part of the body tears it down — in other 
words, gradually wears it out, consumes its tissues ; and if 
this process is continued too long or faster than new ones 
are formed to take the places of those that have been de- 
stroyed, pain, in form of weariness or fatigue — nature's 
warning — ensues, and is a sign that the safety-point has 
been passed and that rebuilding is necessary. 

10. Rebuilding of brain requires more time than that of 
muscle or of bone ; brain-workers, therefore, demand more 
sleep than those who chiefly exercise their muscles. But 
sleep should be sound, unbroken. Students, when per- 
mitted to do so, frequently study at hours of the night when 
they should sleep, and afterwards try to sleep when, from 
exhaustion or worry, they are unable to do so. But wake- 



10 Science and Art of Education. 

fulness, when the physical system needs repairs through 
rest, implies some kind of functional disorder, and un- 
doubtedly is a premonition of brain-exhaustion. Dr. J. L. 
Corning, an authority on cerebral diseases, says : " Derange- 
ment in the function of sleep is an infallible sign that the 
proper relation between waste and repair of brain-tissue no 
longer exists; and that, unless the undue expenditure of 
brain-force be made to cease, cerebral bankruptcy is im- 
pending. * ;fe * The injury produced upon the thought 
and emotional centres by a high degree of worry, conjoined 
with undue intellection, it is almost impossible to over- 
estimate; indeed, a very large percentage of cases of brain- 
exhaustion is directly traceable to this baneful combination 
of causes." 

11. When sleeplessness, headache, loss of appetite, languor 
on rising in the morning, and general debility manifest them- 
selves, and drugging begins, the danger-signals are clearly in 
view, and, if not heeded, a wreck is sure to follow. (For a 
fuller treatment on securing and preserving health, see 
Walker's Physiology and Martin's The Human Body.) 



pan m< 

THE MIND. 



Chapter I.— The Intellect. 

12. Method of Studying It.— What mind is we do not 
know, in fact cannot know ; for it is not a material thing, a 
thing that can be cognized by the senses. Like electricity 
and magnetism, it can be studied only through its manifesta- 
tions or effects. This method of learning is that of in- 
ference, inferring causes from their effects — learning the 
promptings of the mind by the appearances and muscular 
activities that follow them, and that seem to be their effects 
or that we infer to be their effects. Mental states and 
activities must therefore be studied inductively. Generaliza- 
tions must be cautiously made, and no conclusion hastily 
drawn. Facts, sufficient in number, must be observed, 
noted, and classified, before a valid conclusion can safely 
be reached. Nor will one person's facts serve any purpose 
for those of another ; each must make his own observations, 
classifications, and draw his own conclusions. This is what 
each must learn to do in order to become a student of 
mental phenomena. Neither the teacher nor any one else 
can do it for him ; all the help that any one can give him is 
to show him how to study himself and others, and how to 
test his conclusion. 

13. The Body the Servant of the Mind.— The mind 
manifests itself through the body, performs all its observable 



Science and Art of Education. 



or visible work through it ; and one of the objects of educa- 
tion is the training of the body to become the obedient and 
skilful servant of the mind. 

Remark. — Under no circumstances must the mind become 
the mere servant of the body, or the body be allowed complete 
control of the mind. 

14. Forms of the Material World and How Cognized. 

— The material world presents itself to us in six different 
forms: i. Colors and figures ; 2. Sounds ; 3. Heat and cold ; 4. 
Hardness and softness, smoothness and roughness ; 5. Tastes ; 
6. Smells. And the organs through which these forms are 
cognized are the senses, the feelers — seeing, hearing, touch- 
ing, tasting, and smelling. It is, however, the mind that 
sees, hears, touches, tastes, and smells, and not the organs 
which it uses to do so. 

15. When an object affects a sense-organ, the nerves con- 
nected with the organ convey the impression to the mind ; 
in other words, make the mind conscious of it. If the 
object is of sufficient interest or importance at the time, the 
mind observes it and takes, so to speak, an impression or 
image of it. This impression is termed a percept. 

Remark. — A percept remains before the mind, or in con- 
sciousness, only as long as the object which gives rise to it 
affects the sense-organ. 

16. Attention. — In order that the mind may be in the 
most favorable state for the reception of impressions, certain 
conditions are necessary. One of the most important of 
these is attention, or the concentration of the mind upon 
the object examined or subject studied. 

17. For educational purposes, attention may be divided 
into two kinds, apparent and real ; the former being only 
the appearance of attention, the latter the reality. 

18. Real attention may be either attracted or directed, 
and divided or undivided. 



The Intellect. 13 



Remark.— Attracted attention is also called non-voluntary 
(without an effort of the will), and directed, voluntary (by an 
effort of the will). 

19. Apparent and divided attention, having no value in 
education, need no further consideration here. 

20. Attracted attention is that which is given from interest 
or novelty in the subject examined or studied, and depends 
upon the teacher's ability to make the subject of instruction 
interesting or attractive. 

Remark. — Attracted attention is the only kind that can be 
expected from children. 

21. Directed, or voluntary, attention is that which comes 
from an effort of the person giving it. This kind can be 
expected from persons of sufficient age and judgment to 
appreciate the value of subjects of study, but not from 
children. Voluntary attention must, however, ultimately 
change to non-voluntary. If it fails to do this, the teaching 
is defective — a failure. 

22. Attention cannot be given equally well under all cir- 
cumstances. Its most successful efforts depend upon 
certain conditions ; the most important of which are : (i) 
good health, (2) good light, (3) pure air, (4) proper temper- 
ature, (5) comfortable seats, (6) absence of distracting 
objects, (7) proper position, and (8) close classification. 

23. Conditions of attention, however important, do not 
imply attention ; they imply simply that everything neces- 
sary, so far as the pupils and their surroundings are con- 
cerned, has been supplied. The next thing to do is to 
secure attention, and this not all teachers can do with equal 
success. As a requisite, to begin with, the teacher must 
have the confidence of his pupils. He should be : (i) master 
of the subjects he teaches; (2) unhampered by the use of a 
text-book during recitation; (3) cheerful; (4) interested; 
(5) in earnest. He should begin where the children's knowl- 
edge ends ; and should excite curiosity. 



14 Science and Art of Education. 

Remark. — The teacher should not abruptly pass from one 
subject to another, but should by gradual steps lead his pupils 
to it. Inattention is frequently caused by abrupt transitions. 

24. Attention must not only be secured, but must be 
continued or held. The following suggestions, if carried 
out in the proper spirit, not merely in a mechanical manner 
will aid the teacher in keeping the attention of his pupils : 
I. Do not attempt to teach anything that is beyond the 
ability of the pupils ; 2. Keep curiosity aroused ; 3. Let the 
age of the pupils determine the length of the lesson ; 4. Let 
the advancement determine the subject and the lesson ; 
5. In teaching children, sense-wholes should be taught be- 
fore their parts ; 6. Go from the known to its related un- 
known ; 7. As a general rule, tell nothing to pupils which 
they can find out themselves or be led to find out ; 8. Vary 
your mode of instruction, your mode of presenting subjects ; 
9. As far as possible, give every pupil a share in the reci- 
tation ; 10. Use illustrations and apparatus; 11. Do not 
teach longer than you have the attention of your pupils ; 
12. Utilize the instincts — activity, curiosity, imitation, etc. 

Remark on Condition 10 of the Foregoing. — If you 
have apparatus, teach with it. Requiring pupils to prepare 
lessons frotn imperfect descriptions or poorly-made illustrations 
or pictures, when a school has the apparatus with which the 
subject may be studied, is inexcusable. 

Remark on Securing and Holding Attention.— To 
secure and to hold attention, something new must be presented, 
or the mode of presentation must be new. No one can give 
continued attention to that which yields nothing. 

25. Like the other powers of the mind, attention may 
be cultivated. The following are some of the means that 
may be used for this purpose : i. Read or relate something 
of interest and value to the pupils, to be reproduced by 
them ; 2. Require them to reproduce from memory a con- 
versation, lecture, address ; 3. Require the reproduction of 
a problem read to them or by them ; 4. Require the repra- 
duction of a paragraph, article, or selection read by them. ; 



The Intellect. 



Remark. — Special periods for the purpose of cultivating the 
power of attention are not to be recommended ; for the same 
ends may generally be attained in the regular classes with the 
daily work. 

26. Attention Depending upon Pupils* State of Mind. 

— Besides the foregoing conditions of attention, there are 
others which, on account of their relation to successful 
school-work, are of sufificient importance to merit separate 
consideration, i. The pupils may be tired ; 2. Their minds 
may be occupied with something to them of more interest 
than that which the teacher is endeavoring to introduce ; 3. 
They may be in a so-called neutral or indifferent state ; 4. 
They may be in a state of expectancy — waiting, with inter- 
est, for the lesson to be commenced. 

In the first case, they should either be excused from fur- 
ther work, or given some such exercises as the making of 
forms of different kinds of material ; work in which they 
can be interested and which will not tire them. 

The second state may be one of antagonism or opposi- 
tion, one in which they refuse to take part or interest in 
the subject which the teacher desires to introduce. In this 
case the only remedy is to prepare the way, step by step, 
by the introduction of something which shall cause them to 
forget their former thoughts and thus prepare them for the 
introduction of the lesson for the period. 

The third condition needs the same kind of preparatory 
treatment as the second, the only difference being that, as 
a general thing, it requires less effort on the part of the 
teacher. 

In the fourth condition the pupils are eagerly waiting for 
the lesson to begin, and consequently need no preparation 
for it. 

27. Perception— Observation — Reflection.— As already 
stated, the impression which an object makes upon the 
mind through the senses is called a percept ; in other words, 



1 6 Science and Art of Education. 

a percept is a mental construction of an external object — a 
construction in the mind by the mind itself. 

A percept is a product, and the operation or act that 
gives rise to it is perception. 

Percepts are of two kinds, original and acquired. 

Remark i. — Some psychologists apply the term perception 
to a complete mental picture of an external object, and percept 
to a single element of it obtained through one of the senses. 

Remark 2. — Perceptive knowledge, or that obtained from 
the direct or immediate apprehension of an object, is also called 
presentative knowledge, and its revival or representation from 
memory, representative knowledge. 

28. Our knowledge begins with experience ; it depends, 
consequently, upon the completeness and clearness of our 
percepts ; and the quality of our percepts depends upon 
observation — upon attention to all the particulars or char- 
acteristics of an object. Careful observation therefore lies 
at the foundation of education. 

29. Not only can the mind observe the outer, the world 
of matter, but it can also turn itself in upon itself, so to 
speak, and scan its own states and activities. This process 
— observing what the mind itself is doing — is called intro- 
spection, reflection ; sometimes, internal perception. 

30. Memory. — Memory (figuratively speaking) is the 
mind's treasure-house, the power of preserving and recalling 
the facts of consciousness. When attention is withdrawn 
from an object, its percept passes from consciousness to 
the memory. 

31. For the convenience of study, memory may be con- 
sidered under two distinct heads, retention and recollection. 
Retention is the mind's power of holding or preserving the 
facts of consciousness, and recollection that of bringing 
them forth when they are wanted. 

Remark. — James Mark Baldwin, in his Elements of Psychol- 
ogy, says : 

*' In considering the entire mental function which we call 



The Intellect. 17 



memory, we find that it involves several factors or stages, 
which are sometimes treated as distinct operations, but may 
properly be considered, as we find them, together. Together 
they constitute a chain of events whereby the mental life of the 
past is retained and utilized in the present. First, there is the 
permanent possibility of the revival of a past experience when 
its first circumstances are repeated; this is called Retention. 
Next, there is the actual return of the image to consciousness : 
Reprodnctto7i. Third, this image is known as having already 
been presented in our past experience : Recog7iition. And 
finally there is, in most cases, an immediate reference to the 
exact past time of its first experience : Localizati07t in time. 
These, taken together, constitute a finished act of memory. 
Accordingly, memory may be defined as a mental revival of a 
conscious experie?ice'' 

32. As before remarked, when an object of perception 
no longer affects or stimulates a sense-organ, no percept of 
it is in consciousness ; recollection therefore does not bring 
percepts before the mind, but transcripts, mental represen- 
tations, images, ideas, or concepts of them. 

Remark i.— The term concept is preferable to that of idea 
for recalled mental pictures, because idea is used in so many 
senses as to lead to confusion in the minds of students of men- 
tal phenomena. Trench says : " The word idea is, perhaps, the 
worst case in the English language. One person, for example, 
has an idea that the train has started, another had no idea that 
the dinner was so bad." 

Remark 2. — Conception is a constructing process, and its 
products are concepts. 

Remark 3. — All forms in which past experiences may be re- 
called and brought before the mind will in these pages be con- 
sidered as representations or mental pictures, and all mental 
pictures, except percepts, as concepts. 

33. The mind's power to retain facts depends greatly 
upon the state or condition in which it is when it receives 
them. Among the most important conditions of retention 
are the following : i. Healthy and fresh state of body and 
mind ; 2. Undivided attention ; 3. Thorough, clear, and 
distinct comprehension ; 4. Lively and sincere interest ; 5. 
Determination, or force of will ; 6. Repetition ; 7. Suffi- 
cient time for the impression to be made. 



1 8 Science and Art of Education. 

34 An examination of the foregoing conditions of reten- 
tion reveals the fact that retention is not an active power 
of the mind, and that it is cultivated only through the ac- 
tivity of the other powers ; its highest degree of success 
depends, therefore, upon the perfection of the activity of the 
other powers. 

35. Mental images play an important part in remember- 
ing. Kay, in his book on the memory (page 208, etc.), says : 

" The subject of mental images is one that has hitherto re- 
ceived but little attention, and yet it is one of the deepest in- 
terest, and calculated to throw light upon many obscure mental 
phenomena. Whenever a sensation or an idea is presented to 
the mind, a mental image or conception of it must be formed 
in order to its being perceived or understood. In proportion 
to the clearness and distinctness of the image will be the under- 
standing of it by the mind, and the hold taken of it by the 
memory. 

*' As there are different kinds of sensations and different 
classes of ideas, there exists a like variety among mental 
images ; and some minds excel in some, others in other. Thus, 
some may excel in the formation of visual images, others of 
auditory ones. The former will remember best those things 
that are presented to the eye, and of which they can form 
visual images ; the latter, such as are addressed to the ear, and 
form auditory ones. The former will take in and remember 
what they read, the latter what they hear ; the one will learn a 
language most easily by the eye, from books ; the other by the 
ear, from conversation. Some, in listening to a discourse, 
image every word they hear as it appears to the eye ; while 
others, with the auditory faculty largely developed, will image 
what they rdUd as if it were addressed to the ear. Others, 
again, in reading or in listening to a discourse, will attend only 
to the sense or meaning, and form sense-images. These can 
give the substance of what they have read or heard with great 
accuracy, though they may not perhaps be able to recall any of 
the words. In each case it is of importance to ascertain in 
what direction the image-forming power of the mind chiefly 
lies. 

" Further, not only are there images of the eye and ear, and 
of the. other senses, but there are also images of muscular 
movements, as of the tongue and hand. Some may not re- 
member much of what they see or hear, but remember readily 
what they say or do. Hence some children learn best by re- 
peating aloud, others by writing down what they wish to remem- 



The Intellect. 19 



ber. Most persons have probably observed, in writing a word 
in regard to the spelling of which they are sometimes in doubt, 
that if they write it at once, without thinking about it, they 
usually spell it correctly; but if they doubt and hesitate and 
think, they become uncertain, and most probably spell It wrong. 
The reason is that the mental image which directs the hand is, 
in this instance, a surer guide than that furnished by the intel- 
lect. In such cases the more the mind is engaged in thought 
the less able is it to listen to those inner promptings of our na- 
ture — the muscular images of past movements, on which so 
much that is finest and most delicate in our action depends. 

" But not only are there in the mind mental images of sensi- 
ble objects, and of muscular movements, of what we feel and 
what we do, but every thought, however abstract or apparently 
disqpnnected from sensible objects, has its image in the mind. 
We can only conceive an abstraction by having an image of it. 
The abstract idea of a triangle, which is not any particular tri- 
angle, but represents the properties common to all triangles, 
has as much its image in the mind as any individual triangle 
that may have been before it. Further, we must regard each 
abstract idea as having a physical state corresponding to it; 
and hence we can localize abstract ideas and recall the occa- 
sions when they were present." 

36. Not only must facts be retained in the mind for 
future use, but they must be recalled when wanted. Minds 
differ much in their recalling power ; some readily recall 
one kind of facts, others another kind ; but, in either case, 
the ease with which they can be brought forth depends 
upon the manner in which they were stored away in the 
memory. If they are associated, or presented to the mind, 
in a systematic, related manner, they will be returned in 
the same manner, every link in the chain of ideas suggest- 
ing its neighbor. 

37. Concepts or ideas may for educational purposes be 
associated in*four ways : i. By contiguity; 2. By similarity; 
3. By contrast; 4. By cause and effect. Contiguity means 
adjoining, contact ; similarity, likeness, resemblance ; con- 
trast, dissimilarity ; cause, that which produces a change, 
an. effect ; and effect, that which has been brought about 
or produced by a cause. 



20 Science and Art of Education. 

Remark. — Other modes of association besides the foregoing 
are sometimes given by writers on psychology, but as they are 
of no service to teachers, they are here omitted. 

38. The underlying principle of contiguous association 
is that concepts or experiences which occur together or in 
immediate succession afterwards tend to revive one another 
Observation, too, seems to teach that the mind integrates 
or " completes any process upon which it enters, if it has 
performed the same process before." 

39. Contiguity may be in time or in place. One thing 
suggests another that appeared before the mind with it or 
immediately before or after it. Things that were before 
the mind either at the same time or in successive time, or 
that were before it together in space (place), are revived 
together or have a tendency to reappear together. 

Remark. — A number of dissimilar objects may be placed in 
a fixed order and remembered or recalled in that order, by as- 
sociating them in the mind invariably in the same order. 

40. An object or concept will bring before the mind 
others that bear a resemblance or an analogy to it. When 
a relation of some kind can be discovered among objects 
or concepts, and this relation is made the basis of a sys- 
tematic arrangement of the objects or concepts, so that any 
one in the series, in consequence of the relation, will sug- 
gest the next, the recalling process is much easier than 
when the association is arbitrary, mechanical, without re- 
lation. 

41. By the association of similars we are enabled to form 
classes of objects or concepts, on account of their resem- 
blance in form, material, or quality, and to recall them. 
By their resemblances, also, we trace the relations of words 
and thus discover their meanings. The study of languages 
is greatly shortened by the association of similars. 

42. For the purpose of firmly fixing anything in the mind 
and readily recalling it when it is wanted, association by 



The Intellect. 21 



contrast is more serviceable than association by resemblance 
or analogy. Tate, in his Philosophy of Education, says : 
" Associations of resemblance are rarely so vivid as those 
of contrast ; and hence it follows that scenes or events 
which are in contrast with each other are more likely to be 
remembered than those which have a resemblance. Con- 
trast, like light and shadow, makes the objects more promi- 
nent." 

Remark. — Only such things as have some points of similarity 
can be contrasted. Those which cannot be compared cannot 
properly be contrasted. 

43. Landon, in his School Management, says : ** The 
natural relationship which links ideas by means of cause 
and effect renders them eminently suggestive of one 
another ; and where the connection exists, to fix it clearly 
in the mind is one of the most powerful means of associa- 
tion at our disposal. It is especially valuable in lessons 
from the physical or natural sciences ; and should be much 
more generally employed in the teaching of such subjects 
as history than it appears to be." 

Tate, the author above quoted, says : " The minds of 
children are so constituted that they most readily remember 
effects in connection with their causes ; for example, they 
readily associate the light of day with the presence of the 
sun ; storms with winds and clouds ; the heat of summer 
with the long days of sunshine ; the improvement of the 
mind with application to study ; misery with crime, and 
happiness with virtue ; and so on. Associations of this 
kind are most interesting and instructive ; one idea be- 
comes the nucleus of a whole series, and idea becomes 
so linked with idea that we are enabled to form a con- 
tinuous chain of them. Thus we readily remember the 
following chain of associations : Rain falls from the clouds ; 
the clouds are chiefly formed by winds and mountains ; the 
cold on the tops of the mountains condenses the moisture 



a Science and Art of Education. 

in the air, and thus clouds are formed ; the cold on the 
tops of the mountains is caused by the thinness of the air, 
etc. ; thin air is colder than dense air, because it has greater 
capacity for heat, and so on." 

• 44. From the foregoing statements concerning the link- 
ing of concepts it will readily be observed that there are 
really but two forms, or modes, of association, arbitrary or 
mechanical, and philosophical or logical. Association by 
contiguity is arbitrary or mechanical, and the association 
of concepts by such a relationship that one calls into con- 
sciousness another bearing a resemblance of some kind to 
it, or belonging to the same logical connection, is properly 
termed philosophical or logical. 

45. Association by contiguity, of necessity, has its place 
in the acquisition of knowledge; but philosophical or logical 
relationships are more valuable in the development of the 
mind, because they call into activity the higher mental 
powers. No philosophical or logical bond can be dis- 
covered without thought. 

Remark. — Children and the illiterate depend chiefly upon 
mechanical associations ; older people, and especially the edu- 
cated, use philosophical or logical relationships. 

46. There is only one mode of acquiring facility in the 
suggesting, or recalling, process, and that is intelligent, per- 
sistent practice. 

47. The following are some of the subjects that may be 
taught by resemblance and contrast. 

A. Resemblance. — a. Geography. — North and South 
America ; South America and Africa ; California and Spain 
or France ; Italy and India ; Australia and Cuba ; North 
America and Europe ; Pennsylvania and Missouri or West 
Virginia ; New York and Boston or Chicago ; Baltimore 
and Philadelphia ; Richmond and Albany ; Atlantic Ocean 
and Pacific Ocean ; products of East Indies and West 



The Intellect, 23 



Indies ; climate and products of southern U. S. and those 
of India ; etc. 

b. History. — Settlement of Massachusetts and of Virginia, 
of New York and of Massachusetts, of Pennsylvania and of 
Maryland, of Ohio and of Connecticut, of California and of 
Kansas, of New Jersey and of Georgia ; Washington's and 
Jefferson's administrations, Washington's and Lincoln's ; 
Columbus and Captain Cook, Lincoln and Garfield, Grant 
and Napoleon, Bacon and Newton, Pestalozzi and Horace 
Mann, Bryant and Longfellow, Daniel Webster and Jrhn 
C. Calhoun, Horace Greeley and John Bright ; etc. 

B. Contrast. — a. Geography. — The Old World and the 
New ; the two hemispheres (eastern and western); London 
and New York ; Spain and Italy ; coast of Europe and of 
North America ; eastern coast of North America and west- 
ern coast of same ; valley of the Mississippi and that of the 
St. Lawrence ; North and South America ; Africa and 
South America ; Philadelphia and Chicago ; etc. 

b. History. — Settlement of Jamestown and of Massachu- 
setts Bay Colony ; colonists of Virginia and of Massachu- 
setts ; Dutch of New York and Quakers of New Jersey and 
Pennsylvania ; colonists of Maryland and of Virginia ; 
Washington and Jefferson ; Clay and Webster ; etc. 

c. physical Geography. — Surface ot Pennsylvania and of 
New York ; climate of Pennsylvania and of New York or 
Massachusetts ; products of Pennsylvania and of Massa- 
chusetts, Maine, or Virginia ; climate and productions of 
North America and of South America ; valley of the 
Danube and of the Mississippi ; climate and productions of 
Russia and of France, of Ireland and of Australia, of 
Arabia and of India, of Siberia and of British America, of 
Southern Africa and of the southern part of South America, 
of North Temperate and South Temperate Zone ; etc. 

C. Miscellaneous. — Octagon and circle ; degrees of 
hardness and softness ; resemblances of color ; light and 



24 Science and Art of Education. 



heat ; cylinder, cone, and sphere ; forms of letters of the 
alphabet ; spelling of words of similar forms and sounds; 
etc. 

48. The learning of words depends upon association. 
The child associates the name of a word with the form of 
the word, and either one of them suggests the other. So 
also in acquiring knowledge are the names of things asso- 
ciated with the things themselves, and thus remembered 
and recalled. 

Remark. — The work of the primary teacher consists largely 
in helping the children to form permanent associations. 

49. Repetition and reviews constitute the main reliance 
for forming lasting associations ; but the repetitions and 
reviews must not be allowed to become monotonous. Pleas- 
urable emotions, or interest, can be excited and kept up in 
no other way than by variety and novelty. 

50. If memory is found weak in any particular direction, — 
in recalling names, dates, etc., for example, — and any one's 
duties demand much work of this kind, the only remedy is 
daily exercises of the kind required. It is wonderful to 
what an extent the memory can be trained in any special 
direction by persistent efforts. 

61. Sully says : " Committing anything to memory is a 
severe demand on the brain-energies, and should, so far as 
possible, be relegated to the hours of greatest vigor and 
freshness. The morning is the right time for learning. 
In addition to selecting the best time, every resource should 
be used to make the subject as interesting as possible." 

Remark. — A ready memory is undoubtedly an invaluable 
possession, yet it needs to be carefully kept within its legitimate 
bounds ; it should not be permitted to supplant any of the other 
powers — perception, imagination, judgment, etc. 

52. Imagination. — Imagination is the power which forms 
or constructs mental images. It is divided into constructive, 
reconstructive, and productive invention. 



The Intellect. 25 



53. Construction may take place from observation or 
from description — from what we ourselves observe or from 
what we learn from others. Perceiving is a constructing 
process ; it is forming images in the mind ; and though not 
generally called imagination, it deserves as much that name 
as any other picturing process performed by the same 
power. 

54. In the study of all branches of knowledge in which, 
in the absence of the objects or phenomena, mental pictures 
are required, the imagination constructs and paints them. 
In the study of geography and history the use of the con- 
structive imagination is indispensable. 

55. Reconstruction takes place in recollection or from 
memory ; it is rebuilding that which at some previous time 
had been built or formed ; it is constructing according to a 
former pattern or model. 

56. Reconstruction is employed in all recitation. The 
pupil reconstructs his former constructions, and the teacher 
and the class observe whether they are correct or reasonable. 

Remark. — The criticisms made by the teacher and the pupils 
should enable the one who recites to correct and complete his 
constructions. Simply pointing out errors, without indicating 
how they may be corrected, has some value ; but pointing them 
out in such a way as shall enable the student to see them and 
correct them is the proper way to make the corrections. Criti- 
isms should always be helps, not hindrances. 

67. Production, or invention, has reference to original 
combinations or constructions. The ability to make these 
cannot be directly taught ; it can only be encouraged, not 
learned ; it can only be acquired by those who have the 
talents and the patience — by continued study and practice. 
One who succeeds in the highest forms of original construc- 
tion is said to be a genius. 

58. The teacher should not only allow his pupils the 
exercise of originality, but should encourage it whenever it 
is practicable to do so. Throwing them as much as possible 



26 Science and Art of Education. 

upon their own resources is one of the best ways of doing 
this. 

59. Currie, in his Principles and Practice of Common- 
School Education, says : 

" Observation is limited by very narrow boundaries of time 
and space; to whatever extent we pass these, it must be on the 
wings of imagination. Accordingly, descriptions of natural 
scenery and scenes from life, real or ideal, are the field in which 
this mode of intelligence must be exercised ; and both are very 
rich in materials. 

" When the pupil has observed the elements of the landscape 
at home, he is required to carry these abroad and, by modifica- 
tion, interchange, and amplification, to construct another land- 
scape there; the hill, rivulet, meadow, and wood of his own 
native district become the snow-clad peak or volcano, the 
mighty river, the far-spreading desert or cultivated plain, the 
trackless forest, of other lands ; from the summer's heat and 
winter's ice, whose effects he observes at home, he passes to the 
heat of tropical, and the cold of arctic, regions, with the lux- 
uriant vegetation of the one and the stunted growth of the 
other;, the plants and animals of his own country, with their, 
interesting habits, serve as a standard by which he may estimate 
their representatives in other countries; and the notions of the 
adaptations of labor, and the modes of life, which he forms 
from what he sees around him, are drawn upon to construct 
pictures of industry and the habits of other races of his fellow- 
men. Then it is largely by the imagination that a knowledge 
of life is gained, whether of individuals or of communities. The 
life of home or school, and the life of society so far as the child 
sees it, are limited in their incidents ; yet the teacher hesitates 
not to tell or read the story of human life, in its spheres and 
with its diversified enjoyments, in the belief that the pupil's 
experience, narrow as it may be, will enable him to realize the 
emotions portrayed, and gather up the lessons suggested. 
Biography and history are the natural sources of supply for 
materials of this sort; narratives of adventure by sea or land ; 
descriptions of manners and customs; incidents in the life of 
men or societies which embody the virtuous emotions of our 
nature. Ideal life may come in to increase the store of materials ; 
it is equally rich in instruction with the life recorded in history 
Itself." 

60. The happiness of childhood depends almost wholly 
upon the use of the imagination. The plays of children are 
nearly all imitations of the real work of older people, and 



The Intellect 27 



serve as a preparation for the later actual duties of life. 
Thus they make dolls, dress them, feed them, put them to 
bed and rock them to sleep ; they cook, bake, set tables, 
wash dishes, clean h')use, make parties, receive callers, 
build dams, mills, houses, railroads, forts ; ride horses, keep 
store ; act soldiers, doctors, preachers, teachers, etc. 

Remark. — The statement is sometimes made that the imagi- 
nation is more active in childhood than it is at any later period 
of life, but such an opinion rests upon superficial observation 
or investigation. The imagination is not stronger, or more 
active, in early life than it is at a later period ; on the contrary, 
it is weaker, but its activities are more noticeable, because nearly 
all of them are, of necessity, imitations of the actual work of 
grown people. 

61. The use of the imagination is required in every calling 
in life. Nothing can be done intelligently and skilfully 
without a mental picture as a pattern. The mechanic, the 
dressmaker, the baker, the cook, the farmer, the doctor, 
the lawyer, the orator, the preacher, all must use it to meet 
with success in their vocations. 

62, To feel with a person and for a person, it is necessary 
to imagine ourselves in his place. The chief reason why 
some persons seem to be unsympathetic is that their 
imaginative powers are sluggish or dull; they cannot place 
themselves in the position of those who are in sorrow or 
distress. 

63. The imagination is the faculty which constructs forms 
of beauty ; it is therefore one of the powers that give culture 
to taste and refinement. 

64, James Freeman Clarke says : " No man can be wholly 
unhappy who is accustomed to look for beauty in nature 
and in human life. His is a joy which never wearies. 

" All mere drudgery tends to stupefy the imagination ; 
and all work is drudgery which is done mechanically, with 
the hand and not with the mind; when we are not trying to 
do our work as well as possible, but only as well as necessary. 



s8 Science and Art of Education. 

Such work stupefies the ideal faculty, quenches the sense of 
beauty. ' No matter how lowly the labor may be, if a man 
performs it as well as he can, he is an artist.' But when a 
man tries to shirk his work, when he does it in a slovenly 
manner or way, not as well as he might, then he becomes a 
drudge, even though his work be that of a poet or a sculp- 
tor. He ceases to exercise his ideal faculty, and stupefies 
it. Then the sense of beauty dies out of his mind. 

" If men are taught to look for beauty in all that they 
see, to embody it in all that they do, the imagination will 
be both active and healthy. Life will then be neither a 
drudge nor a dream, but will become full of God's life and 
love, and we will be brought into the love of that divine 
beauty which is above all, through all, and in us all." 

65. Explanation of Terms. — i. Analysis consists in 
separating a complex whole, whether material or mental, 
into its elements — the parts of which it is composed. 2. 
Synthesis consists in forming a whole of its parts. 3. Ab- 
straction is mentally withdrawing certain characteristics or 
qualities from objects. It may also be regarded as the 
withdrawing of the mind from all the other properties of 
objects except those under special consideration. 4. Gen- 
eralizing is finding the general or common characteristics 
in a number of objects or concepts. 5. Classification is 
grouping similars into classes, and embraces generalization. 
6. Comparison is simultaneously giving attention to several 
objects to discover their agreements or differences. 7. 
General concept is the term applied to a class, or to a group 
of all the common properties of the objects that compose 
the class. A general concept, therefore, cannot be imag- 
ined, it can only be thought. 8. To apprehend is to seize, 
to take possession of ; to comprehend is to understand. 9. 
A thing is known when it is assigned to its proper class. 
10. Apperception is the appropriation of the new by the 



The Intellect. 29 



old; it is an organizing process. 11. Elaboration is an- 
other name for thinking. 

66. Judging. — When the mind compares two objects or 
concepts directly with each other to determine their rela- 
tion, the process is termed judging, and the result is a 
judgment. The objects compared may both be physical 
(material) or mental, or one may be physical and the other 
mental ; that is, we may compare one object (physical or 
mental) or act with another, or we may compare it with our 
concept of it or of what it should be. The concept or 
image is then taken as the standard, or measure, of the 
comparison. Determining whether a thing is sour, sweet, or 
bitter ; hard or soft ; cold or warm ; black or white ; weak 
or strong ; coarse or fine ; suitable or unsuitable ; good or 
bad ; right or wrong, are all acts of judgment. 

"We judge whenever we affirm or deny one thing of an- 
other. Everything we know, or think we know, involves 
an element of judgment, and, when it becomes distinct 
knowledge, can be explicitly set forth in a proposition. 

"An expressed judgment is a proposition." — Sully. 

67. Without the use of judgment, no advance step can 
be taken in the acquisition of knowledge. Judgment is 
among the earliest powers exercised by children ; every 
act of discrimination requires it — is an act of judging. 
Ideation employs it in completing its images. 

Remark i. — Thinking is a general term applied to discover- 
ing relations ; and its three successive stages are conception, 
judging, and reasoning. 

Remark 2. — The several intellectual processes through which 
the material of knowledge passes from the concrete to the com- 
plete general concept are also usually given as three : i. Com- 
parison ; 2. Abstraction; 3. Generalization. 

Remark 3. — The last step in the conceiving process is that 
of denomination, giving a name to the concept. 

68. Judgments may be explicit or implicit ; that is, we 
may judge consciously or unconsciously. 



30 Science and Art of Education. 

69. Reasoning. — Not all objects or concepts can be di- 
rectly compared with each other. Whenever direct com- 
parison is impossible, a third object or concept must be 
found with which each of the others can be compared and 
their relation determined. 

Remark. — Frequently more than one intermediate term of 
comparison is necessary to determine the relation of the two 
objects or concepts under consideration. 

70. When the relation of two objects or concepts is de- 
termined through on or more intermediate ones, the pro- 
cess is called reasoning. Examples of reasoning : i. Given 
the cost of 4 lbs. of butter, to find the cost of 3 lbs. Here 
the intermediate term, or number, is i lb. 2. To find the 
number of men that can build 80 rods of wall in 16 days, 
if 12 men can build 50 rods in 15 days. Here there are 
two intermediate terms, rods and days ; and the terms of 
comparison are one rod and one day. 3. A house and lot 
costing $6750 were sold at a gain of 12^ percent. ; how 
much was received for them ? Here the term of compari- 
son is I per cent. 4. On butter sold at 40 c. the gain was 
25 per cent. ; what was the cost ? Here the term of com- 
parison is I per cent. 5. Stealing is a violation of law ; a 
violation of law is a crime ; therefore stealing is a crime. 
Here the intermediate term is a violation of law. 6. Per- 
sons who tell falsehoods are not believed when they tell 
the truth ; Alias tells falsehoods ; therefore he is not be- 
lieved when he tells the truth. Here the intermediate 
term \s falsehoods. 

71. Kinds of Reasoning — There are two kinds or modes 
of reasoning, the inductive and the deductive. When a 
general truth is inferred or discovered from the examination 
and comparison of a number of individual facts the process 
is called induction ; and when a particular truth is found 
through a general truth it is called deduction. The appli- 
cation of general principles .to cases that come under them 



The Intellect. 31 



or are included in them is also deduction. When the child 
finds that a hot stove, a gas-flame, a hot coal, etc., burn its 
fingers, and that fire is the general cause of the heat, it 
comes to the conclusion that fire burns ; and it reaches 
this conclusion inductively. Afterwards when the child 
keeps its fingers from hot objects, it does so because it rea- 
sons deductively that all hot objects burn. 

72. All the known laws that govern the material universe 
have been learned by induction. The naturalist cannot 
make nature's laws ; he can only discover them. 

73. Leading pupils to discover their own rules, principles, 
and definitions, from the examination of a number of spe- 
cial cases, is inductive teaching. The deductive method is 
the reverse of the inductive ; by this method pupils are 
taught to apply rules, principles, and definitions. 

Remark. — The inductive is the method of acquiring knowl- 
edge ; the deductive, of using it. 

74. The Quantity of General Concepts.— Concepts, be- 
ing aggregates, or syntheses, of attributes or properties, are 
said to have quantity. By the intensive quantity of a con- 
cept is meant the sum of the qualities which constitute it 
and distinguish it from other classes of concepts ; and by 
the extensive quantity is meant the sum of the concepts 
that may be classed under it. Thus, a quadruped is an 
animal having four feet ; four feet constitute the intensive 
quantity ; and the number of animals having four feet, the 
extensive quantity of the term quadruped. 

75. Formal Reasoning — Every logical form of argument 
(syllogism) or reasoning consists of three propositions, 
judgments, or statements, called, respectively, the major 
premise, the minor premise, and the conclusion. The sub- 
ject and the predicate of a proposition are called its terms ; 
the major term being that of widest scope, or extent, in the 
major premise ; the minor, that of least extent in the minor 
premise ; and the middle, that found in both the others, 



3 2 Science and Art of Education. 

and through which they are compared. The middle term 
includes the minor and is itself included in the major. Ex- 
ample : A is B (is included in B). C is A (is included in 
A) ; therefore C is B (is included in B). Or, taking a 
concrete example : Man is mortal ; Smith is a man ; there- 
fore Smith is mortal. 

76. Systematization. — Arranging concepts or classes of 
objects according to some order or plan is called systema- 
tizing. Arranging the classes or the grades of a school, the 
goods in a store, our daily work, etc., are examples of sys- 
tematizing. 

77. Intuition. — Intuition means immediate beholding. 
We can immediately behold facts of matter and of mind, 
and we can immediately behold, or cognize, necessary and 
universal principles. The senses behold facts of matter 
and of mind ; and therefore give us empirical intuitions, or 
percepts ; the Reason beholds necessary and universal 
principles, and gives us rational intuitions. 

78. Of the correctness of no other knowledge can we 
have less doubt than of our rational intuitions. For exam- 
ple, we can for a certainty know that things which are equal 
to the same thing are equal to each other ; that the whole 
of anything is greater than any one of its parts ; that every 
effect must have a cause ; that no object can exist without 
time or space. 

Remark. — A principle or truth is necessary when we cannot 
think its contrary; it is universal when it is believed and true 
throughout the universe — when it has no exceptions. 

79. Sources of Knowledge.— As has already been shown, 
the mind has three sources of knowledge, perception (em- 
pirical intuition), thought (logical conclusions), and the 
Reason (rational intuitions) 

Remark, — Perception, memory, imagination, comparison, 
abstraction, generalization, conception, judging, reasoning, sys- 
tematizing and rational intuition, belong to the intellect, the 
knowing power. 



Chapter II.— The Feelings. 

80. Numerous divisions of the feelings have been made 
by psychologists ; the following, from Dr. A. Schuyler's 
Empirical and Rational Psychology, are among the simplest. 
I. Physical: sensations, instincts, and appetites. 2. Vital: 
feelings of rest, as fatigue, of vigor or languor, and of 
health or sickness. 3. Psychical : emotions, affections, 
and desires. 

Remark. — Sensations are feelings. A sense is an organ 
through which we feel ; hence the senses are feelers. 

81. Our feelings may also broadly be divided into pleasures 
and pains, or into physical feelings and mental feelings. 
Physical feelings come from affections of the physical or- 
ganism, and mental feelings from cognitions and thoughts. 

Mental feelings, in general, without regard to the usual 
subdivisions, will alone be considered here. 

82. Mental feelings are both causes and effects ; as 
causes they move the mind to action, and as effects (as be- 
fore stated) they result from cognitions and thoughts. 

83. Our pleasures and our pains, our joys and our sor- 
rows, depend upon our feelings ; or, perhaps better, are 
themselves states of feeling. Whatever produces pleasant 
or agreeable feelings is done with comparative ease ; and 
whatever causes painful or unpleasant feelings is wearisome 
and exhaustive. Whatever excites the feelings pleasurably 
is said to be interesting ; and whatever is interesting 
strengthens us and hence aids us in the performance of 
duty. Discouragements, worry, or whatever else depresses 
the feelings, reduces strength and is a hindrance to the 
performance of work, Anticipation or hope of success 

33 



34 Science and Art of Education. 

excites pleasurable feelings and consequently lightens labor. 
Expectant enjoyment as the fruit of success strengthens us 
and enables us to work with ease. 

84. With children, the time that intervenes between the 
performance of duty and its resultant enjoyment must be 
short ; their experiences are too limited to expect much 
enjoyment from that which is far off and, to them, of 
doubtful realization. 

85. That which affords immediate enjoyment is its own 
stimulus to interest and exertion ; but that which gives 
only anticipated, distant realization of success and enjoy- 
ment needs to be invested with novelty and interest to ex- 
cite curiosity — a desire to know — as a condition to lively 
attention and exertion. 

86. The motives that produce pleasurable emotions in 
some natures do not do so in others. The love of praise 
urges some to action ; hope of success, others. A student 
will study hard, day in and day out, early and late, to at- 
tain the end he has in view — prosperity or eminence. A 
man will perform disagreeable as well as severe physical 
labor if it will minister to his ultimate comfort or enjoy- 
ment. It is evident, therefore, that our feelings play an 
important part in the performance of all kinds of work, — 
especially in that of education — and that the motives which 
excite them and urge the student to action should be care- 
fully studied by every one who desires to become an intel- 
ligent and a successful teacher. Garvey, in his Human 
Culture, says : " To interest pupils in their studies is the 
great secret of success in teaching ; but the interest of the 
pupils is best awakened by exciting their curiosity, by hold- 
ing out to them, in a pleasant manner, whatever of strange, 
new, and wonderful the proposed study contains to reward 
their perseverance and attention. * * * The self-sus- 
taining power of pleasurable feeling ought to be a grand 
lever in the hands of the educator. If we once mak^ the 



The Feelings. 35 



subject of study a source of pleasure, we may safely leave 

the mastery of it to the spontaneous energy of the pupil's 

mind. * * * By exciting the emotions we excite the inner 

forces of the mind, which cause it to expand and unfold its 

faculties just as the influence of the seasons excites the 

dormant forces of the plant." 

Remark. — Excitement, as here used, does not mean passion, 
but such interest or pleasurable feeling as shall carry the stu- 
dent through his work with comparative ease, such as shall 
arm him with strength and urge him onward to the reward of 
his labor. 

87. Character. — What a man really is constitutes his 
character ; what others think of him, his reputation. 
Character depends largely upon the feelings ; and since 
character is the man — makes him either worthy or unworthy 
of the confidence and respect of his fellow-men — the moral 
feelings cannot be overlooked in a course of instruction. 

88. The feelings cannot be educated ; they are not think- 
ing powers ; they can only be trained, and they must be so 
trained that right conduct, right and worthy motives, shall 
afford pleasure ; and wrong and unworthy motives and con- 
duct, displeasure and pain. 

89. The character of the teacher has much to do with 
the moulding of that of his pupils. He must be consistent ; 
he must himself be and do what he would have his pupils 
be and do. He must not only hold up for their admiration 
that which is noble and good, but must exemplify it in his 
own daily life. He must show not only displeasure, but 
positive abhorrence, of that which is ignoble, wrong, mean, 
or debasing. Young people, especially children, are imita- 
tive beings, and both consciously and unconsciously acquire 
the habits of those with whom they are frequently associ- 
ated. They are in the presence of the teacher five to six 
hours a day during not fewer than six months of the year, 
and, too, during the most pliable period of their lives ; that 
his influence must be the main factor in shaping their 



^6 Science and Art of Education. 

thoughts and habits must therefore be evident to all who 
properly consider the matter. Hence his character should 
constitute dne of the principal elements of his qualification. 

90. Good conduct comes from good thoughts, and good 
thoughts come from associations and surroundings that in- 
spire theni or prompt them. The mind is ever active ; it 
must have something to act upon. Those who have the 
charge of children must provide this material. If they fail 
to do so, the children will make their own selections ; and, 
owing to their inexperience — their inability to judge cor- 
rectly as to what is best for them — and the natural tenden- 
cies to prefer present and temporary enjoyment, to future 
permanent good, they will, with the rarest exceptions, choose 
that which will in the end prove harmful — destructive of 
success and happiness. 

91. Beautiful surroundings, such as well-arranged and 
well-kept school-grounds, neat and clean school-rooms, 
tastefully decorated walls, are important factors in the cul- 
tivation of good taste and right feelings. Nor must the 
teacher's own appearance be left out of the consideration. 
As has already been stated, but will bear repetition, chil- 
dren are imitative creatures, and both consciously and un- 
consciously adopt many of the habits and traits of those 
who are their daily associates. The teacher's influence 
may therefore justly be considered even greater than that 
of the parents ; for, besides being the instructor of the 
children, during their school days he is continually in their 
presence, and this causes them to be impressed more or 
less firmly not only with his appearance, but even with his 
modes of speech and action. 

92. Children should be trained to conscientiousness ; 
they should be led to take pleasure in that which is right 
and good, and to hate that which is wrong and bad. Lan- 
don, in his School Management, says : "Although the germ 
of the conscience is born within us, it depends almost en- 



The Feelings. ^37 



tirely upon the kind and extent of the moral, religious, and 
intellectual training we undergo as to what strength and 
correctness of action it shall attain. We should therefore 
lose no opportunity of associating emotions of pleasure and 
satisfaction with right doing, of enlightening the children 
as much as possible as to the nature of various actions, of 
strengthening the judgment by suitable exercises, and of 
ennobling the sense of duty. The cultivation of the con- 
science so that it may be a sure guide to the children, and 
that they may readily obey its dictates, should engage the 
teacher's attention throughout the period of school life. 
This training is for the most part indirect, but it is none 
the less important and should be none the less sure." 

93. The feelings cannot decide as to what is right or what 
is wrong ; decisions belong 10 the knowing powers. The 
feelings can only prompt the will to choose and to act in 
accordance with the decisions of the intellect. The feel- 
ings are moved by the intellect, and can be changed alone 
by it. Good thoughts produce good feelings ; bad thoughts, 
bad feelings ; sad thoughts, sad feelings ; pleasant thoughts, 
pleasant feelings ; angry thoughts, angry feelings. Angry 
feelings can be changed to good or pleasant feelings by, 
changing the thoughts that give rise to them to others of a 
good or pleasing nature. 

94. We are responsible for our thoughts and feelings. 
The fact that we can change them proves it. 

95. Strong feeling, excitement, or passion prevents, 
sound, sober thought. It is impossible to reason with an 
angry person. If we desire to reason with him, we must tell 
him an amusing anecdote or something else that will change 
the subject of his thoughts. The same (as before stated) 
applies to pupils in a school. If any of them are angry or 
sulky, they cannot study. The remedy is a change of 
thought, by means of an anecdote, an interesting talk or 
conversation, an illustration, or an experiment. 

Remark. — Conscience includes knowing and feeling. 



Chapter III.— The Will. 

96. The only power of the mind whose operations remain 

to be considered is the will. This is the power that directs, 

chooses, and acts; it is the executive power of the mind. 

It directs the intellect — gives it its work, sets it in motion, 

and waits for its conclusions. If the thing considered by 

the intellect is capable of ministering to our gratification or 

well-being, a desire or longing for it follows. It remains, 

then, for the will to be governed by this desire, or to refuse 

to do so, in view of other considerations. 

Remark. — One of the distinguishing characteristics of will 
is a co7iscious effort to attain a known end. 

97. The manifestations of will may be classed under two 
heads, untrained and trained; in other words, brute and 
rational. An untrained will is one that is governed by im- 
pulse or passion. A person under the influence of such a 
will disregards the light of judgment and reason, and is 
therefore said to act contrary to reason, to be unreasonable, 
stubborn, obstinate, headstrong, ungovernable. This state 
of will is that of children and of the uncultured — an evi- 
dence of an untrained mind. A rational will is one that is 
not hasty, waits for the decisions of judgment and reason, 
and is governed by the highest good of the individual. 
Such a will is said to be strong when it inflexibly ministers 
to the best interests of the person, as dictated by the de- 
cisions of the intellect. It is weak when it permits itself 
to be controlled by the appetites or by any unworthy 
motives. The glutton, the tobacco smoker and chewer, the 
opium-eater, the inebriate, etc., have weak rational wills. 
Like children, they are under the domination of their ap- 

38 



The Will. 39 



petites, and the longer they permit themselves to be thus 
ruled the weaker the will becomes. 

98. When our appetites and our passions gain the mastery- 
over us, they become tyrants, dethrone the will, and reduce 
us to a state of imbecility — pitiable objects ! No one need, 
however, be under the power of his appetite or controlled 
by any debasing motives. The will, like other powers, can 
be trained; but the training must begin early and continue 
as long as may be necessary for the individual to acquire 
self-control. 

99. One of the first things a child should be taught is un- 
questionable obedience to those in authority over it. Besides 
obedience, neatness and cleanliness of person, truthful- 
ness, self-respect, respect for the rights of others, politeness, 
and kindness should early be impressed upon its mind. 
Permitting a child to have its own way is no kindness to it. 
On the contrary, it is doing it a lasting harm. The will is 
trained by rigidly requiring the right to be done and the 
wrong to be shunned. In this way right-doing becomes a 
habit, and this habit is a right will. 

100. Since the feelings urge the will, the importance of 
the formation of right feelings or desires becomes appar- 
ent; for if our desires are in harmony with our best judg- 
ment, and their force is of sufficient strength to move the 
will in the same direction, we have our powers under perfect 
discipline or control. 

101. The will is simply the person willing or exercising 
the power of willing. He is at liberty to select' and to do 
that which will minister to his well-being, either physical or 
spiritual, or that which will prove destructive of it. The 
will, therefore, is an important factor in the formation of 
character. 

102. The power to will, to select one of several duties, 
acts, or things, implies the ability or freedom to reject any 
or all of them, and consequently makes us responsible for 



46 Science and Art of Education. 

our acts. Responsibility does not, however, apply only to 
overt acts, but even to thoughts. This is evident from the 
fact that the will can change the subject of thought, or can 
direct what it shall be. 

103. Since good conduct comes from good thoughts and 
bad conduct from bad thoughts, the importance of having 
the mind occupied with wholesome thoughts is too plain to 
require further proof. 

Remark. — Here, again, the importance of beautiful surround- 
ings, suitable associates, and wholesome literature or reading- 
matter becomes evident. 

104. Bad thoughts can be expelled from the mind only 
by the introduction of good ones — the good will drive out 
the bad, and similarly the bad will drive out the good. 
Here, again, we see that we are responsible for our thoughts. 

105. As before stated, the will and the feelings depend 
upon the intellect. The feelings are excited by the thoughts, 
and the will is moved by the feelings. The feelings of all 
persons are not, however, excited with equal ease. Some 
are moved very quickly, others very slowly, and between 
these extremes many grades are found. Those who are 
easily or quickly excited are said to be passionate, impul- 
sive; they act without proper consideration, without waiting 
for a complete report or decision from the intellect. Rash 
or impulsive persons are neither safe counsellors nor safe 
guides; for they frequently do things of which they after- 
wards are ashamed, and for which they are sorry. 

106. In addition to the foregoing statements concerning 
the importance of training the will to depend upon the in- 
tellect for its course of action, it should be emphasized that 
this dependence must be formed into a habit. 

107. Habit. — Habits control nearly everything we do, 
and it is only after we can do a thing from habit that we 
can do it with ease and pleasure. The use of good lan- 
guage, all mechanical executions, thinking good thoughts, 



The Will. \\ 



controlling our feelings, and acting from pure motives must 
largely become matters of habit. Teachers must invariably 
insist upon such conduct from their pupils as is reasonable 
and right ; and this course must be continued until right- 
doing has grown into a habit — until the pupils can be left 
to govern themselves. 

108. HoAvever necessary outward means of government 
may be during the early periods of life, the fact must not 
be overlooked that they effect no radical change in the dis- 
position : they serve simply as a restraint. A change of 
disposition must come from within, and must give stability 
to habits of doing right. Hence the further pupils advance 
in intellectual and moral culture — in their ability to take 
care of themselves — the less care need be exercised over 
them by parents and teachers. 

109. It cannot be too deeply impressed upon the minds 
of teachers that the best book upon psycholog)^ is the liv- 
ing, acting child. Child study, or experimental psychology, 
is bearing fruit for the teacher's guidance that speculative 
psychology never dreamed of. 

110. Teachers who desire to read, more to aid them in 
their studies than is contained in the foregoing notes 
would do well to consult Dr. W. O. Krohn's " Practical 
Lessons in Psychology " and Prof. William James' " Briefer 
Course in Psychology." Those who wish to acquaint them- 
selves with the Herbartian psychology and pedagogy should 
read Lange's *' Apperception " (edited by Dr. Charles De 
Garmo), Dr. Charles A. McMurry's " General Method," or 
Rein's "Outlines of Pedagogics." 



iMPORTANt Observations and Inferences. 

1. Education begins and ends with life. 

2. The object of scholastic education is to teach the 
pupil how to learn. Its methods should therefore be such 
as will enable him to carry on his own education pleas- 
antly and successfully. 

3. Education cannot create powers or faculties : it can 
only develop those that exist. 

4. The mind can develop what is in itself only by its own 
activity ; self -activity (exercise), therefore, educates. 

6, The powers of the child can be exercised by no one 
but the child itself ; consequently, not what the teacher 
does for the child, but what it does itself, educates it. 

6. The inner promptings of the child make themselves 
known by outer manifestations. 

7. The child indicates its own mode of receiving instruc- 
tion: hence the teacher learns from the child how to teach it. 

8. It is natural for a child to seek knowledge. 

9. Gratifying a child's desire for knowledge stimulates 
that desire. 

10. Knowledge begins with experience ; the concrete 
should therefore precede the abstract — things should pre- 
cede their signs or names. 

11. Facts and phenomena should come before laws and 
principles. 

12. Thought should come before expression. 

13. Each mind has its own rate of growth and develop- 
ment. 

42 



Important Observations a?td Inferences. 43 

14. Substantial learning or attainments cannot proceed 
faster than the mind's rate of growth and development. 

16. Attempting to go too fast, or going too slowly, 
weakens the mind. 

16. There is an intimxate relation between the bod}/ and 
the mind ; work of either reduces the power of the other. 

Remark. — Bain, in " Mind and Body," says : *' The fact is 
now generally admittcrl that thought exhausts the nervous 
substance as surely as walking exhausts the muscles." 

17. Rapidly growing children tire easily ; their rapid 
growth reduces their power of endurance. 

18. " Tlie time to acquire skill in the use of any power, 
mental or physical, is when the power is growing." 

19. " The time to teach a thing is indicated by the arrival 
of the child's interest in it. Children's interests change with 
age. What interests them at an early period does not do so 
at a later." 

20. Attention, memory, and imagination are not so many 
separate entities, but rather necessary accompaniments or 
aids to the other powers. We may have as many memories 
and imaginations as there are classes of things to remember 
and to image, each requiring its own peculiar training. 

21. Before a teacher charges his pupils with inattention, 
lie should ascertain the predominating kind of mental images 
they form, whether visual or auditory. 

22. Poor remembering means in most instances poor 
images and understanding, for which teachers should largely 
hold themselves responsible. 

23. Before the teacher begins to instruct, he should as- 
certain the contents of his pupils'. minds, or he may build 
without a foundation. 

24. The present must grow out of and upon the past. 
What the child has not experienced it cannot image or 
comprehend. 

25. Every percept is made of present and past experi- 



44 Science and Art of Education. 

ences. All our mental activities are performed with accu- 
mulated capital ; that is, every thought we think receives 
the benefit of all our past thinking. That is apperception. 

26. Instruction aims at power and skill ; education at 
character. 

27. Adaptation of the subjects of instruction to the 
pupils' growing needs is the key to success in teaching. 

28. How a subject is taught is more important than what 
is taught. 

29. The teacher's life should be an example for his 
pupils, and his success should be measured by his moral 
influence. 

30. The end of school government should be self-control 
and character. 

Herbartianism. — The five methodical steps which, ac- 
cording to Herbart and his disciples, must be taken in teach- 
ing a lesson : i. " The preparation [analysis] ; 2. The pres- 
entation [synthesis;] 3. The combination [association] ; 4. 
The recapitulation [system]; 5. The application." — Langes 
Apperception, 



pavt 3!©^ 

Object-lessons. 

A. THEIR DESIGN. 

a. The training of the senses — observation. 
d. The gaining of knowledge. 

c. The development of the power of thought. 
{/. The cultivation of expression. 

B. THE PLAN OR METHOD OF A LESSON. 

a. Have a well-defined end in view. 

d. Select a suitable object to be used. 

^. Determine the method or order to be pursued. 
d. Lead the pupils to make their own discoveries. 

Remark. — There is no better way of enlarging the children's 
vocabulary and of increasing their stock of knowledge than by 
means of lessons on objects (things). With such lessons they 
can be taught the names, properties, relations, and parts of ob- 
jects. By " object-lessons," however, is meant lessons with ob- 
jects, not simply about them. The children themselves must 
examine the objects. Knowledge acquired in this way is more 
certain and permanent than that which is obtained at second- 
hand — from books or from teachers. 

Children could profitably spend months upon lessons of 
this kind before receiving instruction of any other character. 

45 



pan ^. 

Penmanship. 

Remark. — The doing of anything is best learned by intelli- 
gently directed practice, and this applies to nothing more 
forcibly than to penmanship. Every one, unless paralyzed or 
deformed, can learn to write a neat hand. The wretched writ- 
ing found in most schools and among many persons otherwise 
claiming to be educated is an unmistakable sign of defective 
teaching. 

1. Suggestions. — Proper' position of the body, correct 
penholding, and the best work the pupils are capable of 
doing, mus^ be insisted upon ; and all their writing, until it is 
as nearly perfect as they can make it, should be considered 
practice in penmanship. 

2. Every letter should invariably have the same form, 
that of the so-called standard letters. 

3. Penholding. — The penholder should be placed be- 
tween the thumb and the first two fingers, so that it may rest 
before the third joint of the forefinger. 

Remark. — The penholders and pencils for children should, 
if possible, be no more than an eighth of an inch in thickness. 

4. The thumb and the fingers should be bent outward so 
as to bring the end of the thumb opposite the first joint of 
the middle finger. 

6. To keep the pen at the proper angle, the thumb should 
be pressed a little under the holder. 

6. The left side of the middle finger should support the 
holder just above the pen, and the forefinger close over the 

bolder, 

46 



Penmanship. 47 



7. The pen should be held as lightly as possible. 

Remark. — The suggestions on penholding apply equally to 
the holding of pencils. 

8. Position of the Body.— The body should be erect- 
not resting on the arm that carries the pen — the head 
slightly inclined forward to see the writing. 

9. The forearm should rest on the muscle in front of 
the elbow and the hand on the nails of the third and fourth 
fingers. The wrist should not rest on the desk or the paper. 

iO. If the hand is so held that the wrist is horizontal, 
both sides at equal distances from the desk, the penholder 
will point to the right shoulder and give the right slant to 
the writing. 

11. To enable the children, in their early efforts, to write 
in straight lines, their tablets should be ruled with base or 
guide lines. 

12. Penmanship should be commenced with copying the 
first lessons in reading, and that a good beginning may be 
made, the teacher should show the children how to hold the 
pencils (or the crayon, if they write upon the blackboard), 
and, when necessary, how to form the letters, sometimes 
guiding their hands. 

13. Height of Letters and Spacing — The propor- 
tionate height of the letters, and proper spacing of letters 
and words, should not be overlooked. 

14. Flourishing. — Until the pupils can write a neat, 
plain hand, no efforts at flourishing should be tolerated. 

16. Charts. — Penmanship charts should be placed upon 
the wall above the teacher's blackboard, where the pupils 
can at all times see the correct forms of the letters. 

Remark. — If the foregoing suggestions on penmanship be 
strictly carried out, neither copy-book nor classes in writing 
will be necessary, and better penmanship, in less than half the 
present usual time, will be the result. 



pan w% 

Primary Reading. 

1. Introductory Remarks. — Before an effort is made 
to give formal instruction to children, either in reading or 
any branch of knowledge, their confidence and good wilj 
must be gained. This may be done by engaging them in 
conversation upon something that is familiar to them and in 
which they take an interest. These conversations should 
be continued until the children's timidity has been overcome 
and they freely converse with the teacher. During these 
familiar talks as much knowledge should be drawn from 
them as possible. In this way, too, the teacher can learn 
the extent of their stock of knowledge, and' this informa- 
tion will serve him as a basis for the superstructure of knowl- 
edge which, under his guidance and superintendence, they 
are to rear. 

2. Reading is thinking — not mere word-calling ; con- 
sequently the more thoughtful and intelligent the pupils are 
made by such preparatory lessons as those on objects, the 
better they will be prepared for reading. 

3. Reading may properly be considered under two heads, 
impression and expression, or silent reading and audible 
reading. Silent reading has for its object the getting of 
thought, and audible reading that of conveying it to others. 
The getting of thought is the first and chief thing to be 
aimed at. When this end has been attained, — the thought 
found, — conveying it to others by means of audible reading 
becomes comparatively easy; for emphasis, pause, and in- 
flection take care of themselves — the thought controls them, 

48 



Primary Reading, 49 



4. Various methods of teaching beginners to read have 
from time to time been advocated and practised. Among 
them may be named the alphabetic, word, phonic, and sen- 
tence. But as no one alone of these methods is free from 
objections, it is found best to form a method that combines 
what is best in all of them. 

6. The word method seems to be the simplest, and there- 
fore the best to begin with. This method follows that of 
nature, presenting wholes before parts. It is also philosoph- 
ical, for, in order to read, to get the thought, words must be 
recognized as wholes, each as a single picture. If either 
the names or the sounds of the letters, instead of the whole 
words, attract the attention of the pupils, they lose the 
thought, and merely pronounce the words. 

Remark. — Since, in order to read, children must recognize 
words, it is not only reasonable, but a saving of time, to begin 
with words. 

6. Words with which the children are familiar should at 
first be taught. The words which they recognize through 
the ear they must now learn to recognize through the eye. 

Remark. — The length of time required to impress a word 
pcimanently upon the mind and to make its name of ready 
recollection depends ilpon the interest with which the te.icher 
presents it. To make this work a success he must have a num- 
ber of devices at ready command. 

Suggestions for Teaching Primary Reading.— i. Select 
a word that is the name of an object in which the children 
take an interest. Engage them in a conversation about the 
object, to interest them, to gain their attention. 

2. Write the word upon the blackboard. Tell them what 
you have written (its name) and urge them to observe it 
carefully, so as to be able to recognize it when they see it 
again. 

3. Erase it and write it among others rpon the black- 
board. Tell them to find it. 



5© Science and Art of Education. 



4. By means of a variety of exercises of this kind and of 
others, impress the word upon their minds, always associat- 
ing the form (written word) with its name. 

5. Test the impression with word-cards, charts, and any 

other means which you may have at your command or 

which you may be able to devise. 

Remark. — Writing the word among others upon the black- 
board affords a good test of the impression. 

6. To make the impression permanent, the children 
should be required or urged to copy their lessons both upon 
the blackboard and into their tablets. The tablets used for 
this purpose should be ruled, and thelineg should be not 
less than three eighths of an inch apart. As before stated, 
the pencils should be very thin — one eighth of an inch in 
thickness — to enable the children to hold them properly. 

7. Every lesson should begin with a review of the pre- 
ceding lesson or lessons. Every word taught should be 
reviewed from day to day until it is permanently impressed 
upon the mind and can instantly be recalled. The recog- 
nition of words should be made automatic. 

8. After the first word has been taught, others should be 
taught to combine with it, or should directly be combined 
with it, to form a phrase or sentence. 

9. As many new words should be taught at each lesson 
as the pupils can well learn. 

10. The teacher should keep a memorandum of every- 
thing given to the class, to be used as material for reviews. 
He should also prepare as many slips of paper with the 
words which the children have learned written upon them, 
as he has pupils. These slips may be kept in pasteboard 
boxes, and whenever the children are at leisure — when 
they have performed their other tasks, or their assigned 
work — they should take out the slips and see how many of 
the words they remember. 

11. The sentences written upon the blackboard as read- 



Primary Reading. 5 1 



ing exercises may also be copied upon slips of paper and 
kept in boxes for the children to read after their other 
work has been performed. 

12. Lists of the words taught may be written upon the 
blackboard, where the children can, from their seats, see 
them. Of these they should be encouraged to make as 
many sentences as they can. The sentences should be 
brought to the class and kindly criticised — errors pointed 
out and improvements suggested. These exercises train 
children to the correct use of language. 

13. As before stated, the words of the reading-lessons 
should at first all be taken from the children's vocabulary, 
or stock of words in use. 

14. The statements, or sentences, should as far as pos- 
sible be about something that interests che children. State- 
ments about themselves frequently prove interesting to them. 

15. The teacher should have a good supply of toy ob- 
jects. At the recitation every member of the class should 
be given one, and the children encouraged or requested to 
talk about them. Their talks, or sentences, may be written 
upon the blackboard and read, or only the most suitable 
ones may be written upon the blackboard and read. 

16. Number-lessons may also be made reading-lessons. 
The statements of facts which the children make may be 
written upon the blackboard and read, or they may write 
them, at their seats, into their tablets, bring them to class, 
and read them. 

17. Lessons on size, form, weight, position, color, quality, 
etc., may be used in the same manner and for the same 
purpose as those on number. 

18. The reading exercises should be varied from day to 
day ; that is, of the same words as many different sentences 
should be made as possible. Varying the exercises not 
only lends interest to the work, but helps to impress the 
words upon the children's min^s, 



52 Science and Art of Education. 

19. The sentences should at first all be short. Long 
sentences are too difficult for beginners ; their length pre- 
vents the children from readily grasping the contained 
thought. 

20. The reading exercises, the words, phrases, and some- 
times sentences, may be taken from a suitable book, a 
Primer or First Reader. 

21. The pupils should at first read only from the black- 
board ; afterwards also their own written work, from their 
tablets, papers, or slates. 

22. After the pupils have learned from one hundred to 
one hundred and fifty words by sight, by the word method, 
they should be taught the analysis of words into their 
sounds. They should be shown that the names of all 
words are made of combinations of sounds. This may be 
done by taking suitable words and pronouncing them slower 
and slower each succeeding time until the separate sounds 
are heard. Words in which all the sounds may be pro- 
longed should at first be used. 

Remark. — Some teachers introduce the sounds of the let- 
ters and their signs almost from the first lesson in reading, and 
do it successfully. 

23. The analysis of words into their sounds should be 

followed by the synthesis of the sounds. The children 

should be made acquainted with the letters that stand for 

the sounds, and should have practice in determining the 

pronunciation of words. 

Note. — Teachers who desire a carefully worked-out system 
of primary reading, in general accord with the foregoing, will 
find '• The Rational Method," published by Silver, Burdett & 
Co., New York, the most recent and the best. It is a thought- 
method, and the shortest to the mastery of words and intelli- 
gent reading. 

24. All words that cannot well be taught by sounds 
should be taught by sight, or as wholes. 

25. The English language contains about forty-three 



Primary Reading. 



S3 



different sounds, and has only twenty-six letters by which 
to represent them ; some letters, therefore, stand for more 
than one sound. The particular sound for which a letter 
stands is indicated by a sign placed upon the letter, above 
it or below it. These signs are called diacritical marks. 

26. Whenever a letter is taught as the representative of 

a sound, its mark should be taught in connection with it. 

Remark. — The diacritical marks enable the children to dis- 
cover the pronunciation of words. 

27. Whenever a new word is introduced, its sounds 
should be indicated by their proper marks. In the dic- 
tionaries the pronunciation of words is indicated by dia- 
critical marks. In some of the first reading-books diacriti- 
cal marks are also used. 

Sounds of the vowels and consonants as given by Webster : 



Vowels. 



a as in 


,ape 


b 


as in note 


a '• " 


at 


6 




" not 


H " " 


Srm 


5 




" ^r 


4 " " 


ask 


6 




" love 


a " " 


are 


P. 




" mqve 


§ " " 


all 


o 




" WQlf 


9 « " 


what 


o 




" obey 


a " " 


senate 


oS 




" moor. 


e " '• 


me 


do 




" book 


e " " 


met 


u 




<* mute 


3 « « 


there 


ti 




" tJp 


^ « « 


t^rm 


u 




" (uH 


£ " " 


thgy 


u 




" rule 


e " " 


event 


(J 




" b1arn 


1 « It 
T " " 
T " " 
^ <i «< 


Tee 
It 

firm 
pTque 


I 




•' any 
" hymn 
" myrrh 




Consonants. 






•€ as in -eat 


a 


as 


in fiQger 


5 " . 


* jent 


s 


« 


" is 


■ch " • 


* -ehorus 


th' 


« 


" these 


g " • 


' germ 


I 


« 


" exist 


g - " to 









54 Science and Art of Education. 

28. As a further aid to pupils in discovering the pro- 
nunciation of words, silent letters may be marked, also 
double vowels and double consonants. 

29. The names of the letters of the alphabet afford no 

aid to the children in discovering the pronunciation of 

words ; they serve merely to designate the letters when 

speaking of them, and should therefore be taught only 

indirectly or incidentally. 

Remark. — Oral spelling cannot be substituted for real spell- 
ling, and if practised at all, should be postponed until it will 
not interfere with reading. 

30. To familiarize the pupils with the forms of words 
and to afford practice in forming them and in penmanship, 
they should be required to copy carefully and neatly, into 
their tablets, either a part or the whole of every reading- 
lesson until they have completed the Second Reader ; and 
the tablets or books should be preserved to note the prog- 
ress of the pupils in penmanship, spelling, etc. 

31. All new words and words not well known should be 
written upon the blackboard, and the pupils drilled upon 
them until they can readily name them. They should not 
be permitted to attempt the reading of a sentence until they 
can, upon sight, pronounce every word in it. 

32. Instead of being asked to read a sentence or para- 
graph, they should be requested to tell what it says. They 
should generally tell it in their own language. 

33. Conversational tones should be insisted upon in 
reading. 

34. If a lesson contains anything worth remembering, 

the pupils should (with some exceptions) be required to 

give the substance of it in their own words. An exercise 

of this kind cultivates their memory, imagination, and the 

power of expression. 

Remark. —Sometimes, too, they may be required to substi- 
tute synonymous words and expressions for those found in 
their lessons. 



Priniary Reading, 55 



35. Words and sentences may be dictated to the pupils 
to be written into their tablets or upon the blackboard, to 
test their orthography and to impress the correct word- 
forms upon their minds. 

36. Owing to the similarity of the forms of letters, the 
change from script to print is not difficult. To make the 
transition, the teacher should place exercises in both forms 
of letters upon the blackboard and let the children compare 
them. A few days* comparison and practice in reading 
print will remove all difficulties in reading the latter. 

37. As soon as the children can, with considerable ease, 
read from the blackboard, they should be allowed to read 
either from slips or cards printed for the purpose or from 
books ; and until they can read fluently from either, they 
should have only one or two pages of reading-matter given 
them at a time. If they read from books, instead of slips 
or cards, they should be given them only during the recita- 
tion. 

Remark. — If the foregoing suggestions with reference to 
reading from slips or cards and books be carefully observed, the 
children will every two or three days receive something new to 
read, and the expectation of this will keep up their interest in 
reading. 

38. The sentences should at first be commenced with 

small letters, and each letter should invariably have the 

same form — that of the so-called standard letters. 

Remark. — The teacher cannot be too careful with his own 
penmanship. As it is to be an example, a guide, to his pupils, 
it should as nearly as possible be perfect. 

39. As soon as it is believed not to confuse the pupils, 
the capital letters should be introduced to begin sentences. 

40. The pupils may read three, four, or more months only 
from the blackboard and their tablets or papers. 

41. Only one thing should be introduced at a time, and 
the teacher should be sure that it is both understood and 
remembered before he introduces anything else. 



56 Science and Art of Education. 

42. Nothing less than the best work the pupils are capa- 
ble of preparing should be accepted from them. Accepting 
carelessly prepared work tends to the formation of careless 
habits. 

43. All the teacher's work should be an example for his 
pupils. 

44. Articulation needs careful attention in every exercise 
in which the voice is used. 

45. After the pupils are far enough advanced to do so, 
they may be required to find all the possible meanings of 
which a sentence permits by emphasizing in succession the 
different words in it. 

46. The children should as early as possible read for in- 
formation, so that a desire for reading may be formed. 
Much of the reading done in the majority of schools adds 
nothing to the pupils' stock of knowledge, neither possesses 
any interest for them, and consequently, in a correct peda- 
gogic sense, does more harm than good. 



Advanced Reading. 

Remark i. — Primary reading, as the term is used in these 
Notes, refers to that done on the blackboard and in the " First 
Reader." 

Remark 2. — If reading is well taught in the primary grade or 
class, that of the higher will largely take care of itself. 

1. The reading-matter for childfen should, as far as pos- 
sible, be of a thought-stimulating, interest-creating charac- 
ter, adapted to their comprehension and power of appre- 
ciation. Much that is now read in the schools falls far 
short of meeting these conditions. 

2. As has already been stated, but will bear repetition, a 
child should not be permitted to read a sentence audibly 
until it has read it silently, and has obtained the thought. 
In other words, it should not be allowed to express the 
thought until it has it. Permitting, or, as is too frequently 
done, requiring, pupils to read, to express thoughts audibly 
before they have any — thoughtless reading — is chiefly re- 
sponsible for the poor reading that is so general in all 
grades of schools. 

3. Thought-getting and thought-expressing are not the 
same. A pupil may make commendable progress in obtain- 
ing thought, but be slow in expressing it. Words are slow 
in impressing themselves upon some minds. There are 
children, too, who learn more readily through the ear (ear- 
minded) than through the eye ; they remember better what 
they hear than what they see. Instead, therefore, of chid- 
ing or scolding them for slowness, the teacher should ascer- 
tain its cause and, as far as possible, provide a reriiedy. 

57 



58 Science and Art of Education. 

4. When pupils are far enough advanced to do so, they 
should, frequently at least, if not generally, be required to 
read a paragraph or selection silently, and then give the 
thought in their own words, orally or in writing. This 
method of teaching reading cultivates both thought and ex- 
pression, and consequently has a greater culture value than 
reading as it is usually taught. 

Remark. — Good or expressive reading should be insisted 
upon in every class or subject in which a pupil is required to 
read. 

• 5. Pronunciation. — {a) Careless or slipshod pronuncia- 
tion seems to be the rule rather than the exception in the 
majority of schools. It is not an uncommon thing to hear 
such pronunciations as the following: Fir iox for^ben iox 
been^ uf for of, sence for sitice^ sepret for separate, evry for 
every, rhetric for rhetoric, generly ior ge?ieraliy, easly for easily, 
practus for practice, calm for calm, kin for can, blessid for 
blessed, goen for going, govner for goz'ernor, histry for history, 
cave for carve, staiim for storm, articlation for articulation, 
peticular iox particular, prinsple iox pri?iciple, artic for arctic, 
ax for acts,plitical iox political, an for and, dag for dog,fourt 
for fourth, you-ri-zon-nears for your eyes and ears, hi-zow-ry- 
zup for his hour is up, tay-cary for take care, gavim for gave 
him, ovis for of his. 

(b) When errors like the foregoing have been hardened 
into habits they are difficult to eradicate ; but they can be 
corrected by persistent daily practice on the vowels and 
consonants, first separately and afterwards in combination. 
A part of every instruction period, as long as necessary, 
should be devoted to phonic drills. 

(c) A reader or speaker can make himself more easily 
understood by a correct and clear enunciation of his vowels 
and a distinct articulation of his consonants than either by 
a high pitch or by loudness. 

6. Emphasis. — (a) Any method of giving special sig- 



Advanced Reading. ^^ 



nificance to an expression is emphasis, and emphasis is the 
life of reading. Correct emphasis helps the hearer to the 
meaning of what is read, and incorrect conceals it, or 
hinders him from getting it. A knowledge of the leading 
principles of emphasis is therefore a necessity to good 
reading. 

(h) Alexander Melville Bell, one of the highest authori- 
ties on expressive reading and speaking, says: " The laws of 
emphasis form a study of the highest intellectual value, 
which has been too little investigated and systematized. 
No other department of elocution can compare with this in 
importance ; yet not only has it been superseded in books 
by unnecessary rules for inflection, and in schools by 
thoughtless imitation, but these rules, and all exercises 
founded on them, constantly violate the laws of (rhythmic) 
accent. Here is one point in which almost absolute uni- 
formity must prevail among all good readers. Set practice 
right in re§pect to emphasis, and inflection cannot go far 
wrong." 

{c) The sense of what is read determines the emphasis to 
the reader, and the emphasis conveys it to the hearer. 

(d) Misplaced emphasis perverts the meaning. Take the 
following, for example, and read it as here indicated by the 
italics, and as it is generally read, and note the sense: " In 
the beginning was the Word, and the Word was with God, 
and the Word was God." This rendering would lead us to 
believe that nothing existed in the beginning but the Word, 
and that at that early period, at least, it was God, but that 
at a later day it was changed to something else. 

Read the foregoing as here indicated, and observe the 
difference: ** In the beginntfig was the Word, and the Word 
was with God, and the JVord was God." 

((?) The fifth verse of the first Psalm is frequently read : 
** The wwgodly are not so ; but are like the chaff which the 
a//«^ driveth away." 



6o Science and Art of Education. 

Without saying anything about the accent on godly in 
^/«^^^/k, this reading permits of at least two meanings: i. 
That there are two kinds of chaff — one driven by the wind, 
which the ungodly resemble; the other not so driven, which 
we may infer the godly are like. 2. That one kind of chaff 
is driven by the wind, the other by something else. 

(/) In Acts 20 : 16 is found a good example to illustrate 
the variety of meanings of which a sentence permits by 
changing the place of emphasis. 

1. Paul had determined to sail by Ephesus.* 

2. Paul had determined to sail by Ephesus. 

3. Paul had deterniiued\.Q sail by Ephesus. 

4. Paul had determined to sail by Ephesus. 

5. Paul had determined to sail by Ephesus. 

6. Paul had determined to sail by Ephesus. 

The first sentence states PauVs determination ; the sec- 
ond, what it /M^been ; the third, that his mind had been 
fully set; the fourth, that he meant to go by water; the sixth, 
the route — by way of Ephesus. 

Remark. — Requiring children, or even more advanced 
pupils, to read all the various meanings of which a sentence 
permits by shifting the place of emphasis, is one of the best 
ways of breaking up monotonous reading. 

For this purpose a sentence like the following may some- 
times be used in connection with their regular reading ex- 
ercises : Did you see Henry this morning ? 

(^) The following are given in some reading-books as 
examples of the use of the circumflex, but the efforts that 
are usually made to read them are in few instances success- 
ful. Properly applied emphasis, however, removes the 
difficulty, sets the inflections right, and expresses the in- 
tended sense. 

A man who is in the daily use of ardent spirits, if he does 
not become a drunkard, is in danger of losing his health and 
character. 



Advanced Reading. 6 1 



Stress upon not makes good sense ; but if placed upon 
drunkard it would make inebriety a necessity to the preser- 
vation of health and character. 

The dog would have died if they had not cut off his 
head. 

Stress upon not expresses the sense, but if placed upon 
both died and head^ we would be told that cutting off his 
head saved his life. 

{h) The kind of emphasis depends upon the kind of 
thought to be expressed. 

7. Pausing. — {a) Pauses, or cessations of the voice, are 
required in reading to enable the hearer the better to grasp 
the meaning of what is read. 

{b) The pauses are governed by the sense and not by the 
punctuation-marks ; hence, though cessations generally 
occur at these marks, they are also required where such 
points would be out of place. 

{c) In the examples which follow pauses are indicated by 
the dash. 

(d) Now — about that time — Herod — the king — stretched 
forth his hands — to vex — certain of the church. i^Acts 
12 : I.) 

The omission of the pause after Herod would signify that 
one of several persons by that name is meant. 

The pause after to vex might be omitted, but its use adds 
force to the expression. 

{e) But — the Jews which believed not, — moved with 
envy, — took unto them certain lewd fellows — of the baser 
sort. {Acts 17:5.) 

Read without a pause after Jews^ conveys the sense that 
those of the Jews who believed not were moved v*^ith envy; 
and with the pause, that none of them believed. A pause 
might also be made after them^ but its omission does not 
add uncertainty to the meaning. 



62 Science and Art of Education. 

(f) There was a man sent from God, — whose name was 
John. {John i : 6.) 

The omission of the pause after God might convey the 
meaning that the man was sent from the God whose name 
was John. 

{g) And — turning the cities of Sodom and Gomorrha into 
ashes — condemned them. (2 Peter 2 : 6.) 

And^ the first word, belongs to condemned (in reading), not 
to turnings and this the pauses serve to show. 

8. Grouping. — {a) Words are spoken either singly or in 
groups. Those that express a unit of thought must, by the 
voice, be grouped into a unit of expression, and read 
throughout, without interruption, in the same tone as if they 
were one word. 

{h) In the following examples the grouped words are 
placed within brackets: 

{c) Forasmuch then as the children are [partakers of 
flesh and blood], he also himself likewise [took part of the 
same]; that through death he might destroy [him that had 
the power of death], that is, the devil. {^Hebrews 2 : 14.) 

Each of the expressions within brackets may be regarded 
as equivalent to a word. 

(a) I charge thee therefore before God, and the Lord 
Jesus Christ, [who shall judge the quick and the dead at his 
coming]; preach the word. (2 Timothy 4 : i, 2.) 

The clause within brackets is subordinate to what follows 
(preach the word), and this must be made manifest in read- 
ing, by grouping and tone of voice. 

{e) [Men shall buy fields for money, and subscribe evi- 
dences, and seal them, and take witnesses in the land of 
Benjamin, and in the places about Jerusalem, and in the 
cities of Judah, and in the cities of the mountains, and in 
the cities of the valley, and in the cities of the south]: for 
I will cause their captivity to return, saith the Lord. 
{^Jeremiah 32 : 44.) 



Advanced Reading, 67^ 



The substance of all within the brackets is, that pros- 
perity shall again return to the land. 

9. Etymology. — Etymology should receive attention in 
connection with reading. 

Remark i. — A good reader reads thoughts, not words. He 
pictures to himself his author's views and expresses them as if 
they were his own. He is unconscious of everything but the 
thought. If pronunciation, inflection, holding of book, posi- 
tion of feet or hands, or anything else, diverts his attention, he 
loses the thought and reads like a machine. 

Remark 2. — Reading cannot be taught by rules ; it requires 
a competent teacher, one who can himself read and teach, and 
who can discriminate between the useful and the useless. 

Teachers who desire to become effective readers, but can- 
not avail themselves of the services of a competent in- 
structor, will find Alexander Melville Bell's " Principles of 
Elocution " an excellent work for self-help. 



^art a^3!9i3I. 



NOTES AND SUGGESTIONS ON TEACHING THE ENG- 
LISH LANGUAGE. 



Chapter I. General Considerations. 

1. Like all other languages, that of the English-speaking 
people conforms to laws, and these laws are laid down, or 
taught, in books on English grammar. 

2. The laws, or principles, of grammar are inductions 
from a study of the mechanism of the language ; and, like 
those of the material universe, are not made, but discov- 
ered. 

3. The laws of language, as found in books on grammar, 
rhetoric, and logic, may be pursued by themselves as sci- 
ences, or they may be introduced as they are needed to en- 
able the student to acquire a correct, an intelligent, and a 
skilful use of the language. 

4. Prof. W. D. Whitney, in his Essentials of English 
Grammar, says : ** It should be a pervading element in the 
whole home and school training of the young, to make 
them use their own tongue with accuracy and force ; and 
along with any special drill directed to this end, some of 
the rudimentary distinctions and rules of grammar are con- 
veniently taught. But that is not the study of grammar, 
and it will not bear the intrusion of much formal grammar 
without being spoiled for its own ends. It is constant use 
and practice, under never-failing watch and correction, that 

64 



General Considerations. 65 

makes good writers and speakers ; the application of direct 
authority is the most efficient corrective. Grammar has its 
part to contribute, but rather in the higher than in the 
lower stages of the work. One must be a somewhat re- 
flective user of the language to amend even here and there 
a point by grammatical reasons ; and no one ever changed 
from a bad speaker to a good one by applying the rules of 
grammar to what he said." 

6. From the foregoing it seems clear that a distinction 
must be made between acquiring the skilful use of a lan- 
guage and the study of its mechanism. The former comes 
from practice, under careful supervision, and is an art ; 
the latter is the result of study, and is a science. The sci- 
ence and the art of language are therefore different things, 
and are pursued for different ends. 

6. The advocates of technical grammar as a general 
school study but a few years ago claimed that it teaches 
its learners to speak and to write the English language cor- 
rectly. When, however, it was asserted by those who had 
given the subject earnest consideration, that it failed to 
prove its claims, its advocates maintained that its short- 
comings must be attributed to the teaching, and not to the 
subject. But this ground having, both by experience as 
well as by a careful study of the subject, been found unten- 
able, has now been abandoned by the foremost thinkers 
and students of education. 

7. Since, therefore, it has been found that the study of 
English grammar does not, in fact cannot, make good writ- 
ers and speakers, much less importance is placed upon it 
than formerly. 

8. Under the topic of Grammar in City Schools, Vol. I 
of the Report of 1888-89, of Dr. W. T. Harris, Commis- 
sioner of Education, has the following in agreement and 
confirmation of the foregoing, viz.: " In the admitted fact 
that the formation of habits of correct speech is not de- 



66 Science and Art of Education. 

pendent upon a knowledge of the rules and definitions rela- 
tive to the construction of language and their mutual de- 
pendence, is found the justification for the lessened weight 
attached to such study." 

The Report quotes the following from Fitch's Lectures 
on Teaching : " The practical art of using language in 
speech or writing with good taste and correctness . . . 
is probably best to be attained by talking to the pupil, by 
taking care that he hears little but good English, by cor- 
recting him when he is wrong, by practising him much in 
writing, and when he makes a mistake, by requiring him to 
write the sentence without one. It will certainly not be 
attained by setting him to learn Murray's, or indeed any 
other grammar." 

The article on the study of grammar closes with the fol- 
lowing important statement, viz.: "The art is the thing 
directly useful : the science has no obvious relation to 
practical affairs. The ability to speak and write correctly 
is not only desirable, but essential in every walk of life ; 
the technical rules of etymology and syntax are almost val- 
ueless in themselves. Therefore, in accordance with the 
movement whose' progress is here recorded, the art, that is, 
the practical, increases in importance in the course of 
study, while the science, that is, the disciplinary, decreases 
in the same proportion." 

9. It is true that formal or technical grammar has some 
disciplinary value, but the amount of it is usually greatly 
overstated, and unfortunately so, for much of this supposed 
value is responsible for the sad condition in which the 
study, if it may be so called, leaves the minds of the chil- 
dren who, reluctantly, spend their time upon it. 

10. Instead of dividing the study of language into gram- 
mar, rhetoric, and logic, and pursuing each as a separate 
subject, all should be taught under the head of language^ 
and each at the proper time made to contribute its share ta 



General Considerations, 67 

■ 1 = — 

the perfection of the whole, namely, the correct, elegant, 
and forcible use of it in writing and speaking. 

Remark. — The last paragraph refers to the work of schools 
below the college. 

11. The use of good language must become a habit — 
'* an unconscious one." But habits are of slow formation ; 
that of language requires years. A correct taste, the ability 
to form a correct judgment of a composition, must be ac- 
quired. Like misspelled words, badly constructed language 
must offend the taste ; that is, its defects or faults must at 
once appear on being seen or heard. No one, therefore, 
whose taste is uncultivated can teach language. He may 
be able to teach grammar, and to some extent rhetoric and 
logic ; but he cannot tell whether discourse, of whatever 
kind, is well constructed, and therefore can be of no ser- 
vice to learners in the formation of taste, or in acquiring 
the use of good language. 

12. From the foregoing considerations the conclusion 
seems evident that language should be taught by practice, 
intelligent practice, upon the principle that we learn to dc> 
a thing by doing it, and not by merely learning about doing 
it. 

Remark. — As before stated, all the technical prragnninr nec- 
essary to a correct, forcible, and intelligent use of English not 
only can be acquired in connection with reading and composi- 
tion, but should be so acquired. 



Chapter II. 

I. Oral Language. 

Remark. — Spelling, penmanship, punctuation, and capitali- 
zation beloncr to composition, and should be taught in coimec- 
tion v/ith it, and not as separate branches of study. 

1. In order that children may talk, they must have 
something to talk about ; that is, they must have a subject 
for conversation. 

2. The subject should be one that will interest them and 
that will add to their stock of knowledge. Stories that add 
nothing to their knowledge, that do not improve their pow- 
ers of observation and thought, or contain anything worth 
remembering, should generally be avoided. 

3. The subject should be one that the children under- 
stand, or can readily be led to understand. It should gen- 
erally be one which they have observed or can observe. It 
should be thoroughly understood by them. Half knowl- 
edge is little better than none at all. 

Remark. — Conversation depends upon observation, memory, 
imagination, and discrimination. 

4. Nothing would be lost, but considerable gained, if 
children had exercises in observation and conversation 
some weeks, or even months, before they begin to read. 

5. All their lessons furnish material for conversation. 
No teacher need be at a loss for material, for he has the 
whole animal, vegetable, and mineral kingdoms from which 
to select. Judgment must, however, be exercised in the 
selection and in the order of presentation. The simplest 
and most interesting to children should come first, and then 

68 



General Co7isi derations. 69 

there should be a regular gradation in the order best 
adapted to the growth and development of their minds. 

6. The most interesting objects for children are living 
things, animals. Those with which they seem best ac- 
quainted, and which they can the more readily examine 
and study, should come first. The cat, dog, cow, horse, 
pig, sheep ; chickens, ducks, geese, turkeys, pigeons, par- 
rots, canary-birds, are not only interesting but profitable 
objects for observation and study. Notwithstanding that 
we seem to be familiar with them, to our discredit it must 
be said, we know little about them. Children should be 
led to study them, and conversations about them will in 
the best way lead to this. 

Remark. — This kind of language work will require the 
teachers to familiarize themselves with the objects which the 
children are studying, and will thus at the same time enlarge 
their own sphere of knowledge. 

7. Flies, bees, wasps, spiders, butterflies, bumblebees, all 
suitable kinds of insects, worms, birds, etc., found in the 
community, afford suitable material for observation, study, 
and conversation. 

8. The vegetable kingdom should also be drawn upon. 
Roots, stems or stalks (or trunks), leaves, flowers, and seeds 
should be observed ; their similarities and dissimilarities 
noticed, and their uses discussed. Plants about home, in- 
cluding grasses and trees, should first receive attention ; 
and it cannot be too deeply impressed upon the minds of 
the teachers that the things themselves, and not pictures or 
descriptions of them, are to be the objects of observation 
and study. Plants may be kept in pots in the windows or 
other places of the school-room. Towards spring, twigs 
may be cut from trees, and, in jars partly filled with water, 
placed in the windows where they will be in the sun, and 
the opening of their buds observed from day to day. 
Seeds may be studied in the fall of the year. Towards 



JO Science and Art of Education. 

spring they may be planted in " shallow boxes, saucers, or 
any other convenient vessels filled with moist earth, sand, 
or saw-dust." They may also be planted on cotton and 
other suitable material floating upon water in wide-necked 
bottles, or they may be put into a sponge placed in a saucer 
of water. In this way their germination, the growth of the 
root, stem, etc., may readily daily be observed and noted. 

9. The various kinds of minerals, including earths and 
stones, can also be made a profitable study, and therefore 
afford good material for conversation. 

Remark. — Jackman's Nature Studies, published by Henry 
Holt & Co., New York, is a book of valuable suggestions. 

10. Conversations may be had about things in the school- 
room, on the school-grounds; appearance of the sky, kinds 
of clouds, direction of winds and their effect upon the 
weather. Things at home, such as may be found in the 
sitting-room, the parlor, on the table, in the kitchen, the 
laundry, the pantry, the cellar, the barn, on the farm, etc., 
may also be used. 

11. The children should have opportunities to learn and 
to tell how the various kinds of business of a community 
are conducted. They should name the kinds of stores ; 
tell what is sold in them ; where and how the goods are 
procured ; in what ways, where, and upon what terms they 
are bought ; also upon what terms generally sold ; and the 
necessity of such places of business. The post-office, ex- 
press-office, bank, court-house, jail, railroad, railroad station, 
should be explained and discussed. Their advantages or 
necessities, as the case may be, should form part of the dis- 
cussion. In the same way may the Sunday-school and the 
church be used and their influence upon the community be 
brought out. The various town and township officers, their 
necessity, duties, and how they are elected, installed, and 
paid, will at the same time be lessons in civil government. 



General Considerations. 7 f 

Remark. — What the children do not know the teachers must 
tell them. 

12. Events of the day that are within the children's 
comprehension, and in which they can be interested, should 
be used for oral language work. 

Remark. — As will have been observed, the foregoing mate- 
rial for conversation is designed to make the children ac- 
quainted with their surroundings, and to make them observant, 
thoughtful, and intelligent. 

II. Written Language. 

Remark i. — All the material suitable for oral language is 
equally so for written work ; and as soon as the pupils can use 
the pen or pencil they should be encouraged to talk with it, or, 
in other words, to write their thoughts, or at least some of 
them. Copying well written extracts and writing from memory, 
are good language exercises, occasionally to be used with be- 
ginners. 

Remark 2. — Besides the regular daily lessons in the various 
branches of study, which should all at the same time be re- 
garded as lessons in language, the following exercises and sug- 
gestions, if intelligently used, will aid the teacher in training 
his pupils to the writing of good English. 

1. Relate something of interest to them and let them re- 
produce it in their own words. 

2. Read short, easy, but complete extracts, stories, or 
items of news, to them, and let them give the substance in 
their own words. 

3. Let them say what they saw on their way to school, to 
the post-office, the store, the church, etc. 

4. Let them write the history of an hour, a day, a week, 
a month, etc. 

5. Let them write a narration of a walk, a ride, a real or 
an imaginary journey, a picnic, party, sociable, or anything 
that has occurred of which they have knowledge, or which 
they imagine as having occurred. 

6. Let them describe a room, house, barn, stable, school- 
house ; the post-office, court-house, church ; a chair, table, 



72 Science and Art of Education. 

bench, door, window, knife, pair of scissors, thimble, lock- 
ing-glass, clock, stove, wash-stand, comb, brush, sewing- 
machine, hat, cap, desk, tree, fence, gate, wagon, wheelbar- 
row, shovel, fork, hammer, broom, horse, cow, dog, cat, rat, 
mouse, pig, sheep, fly, spider, caterpillar, beetle, plant, flower: 
in short, anything that admits of description and that, in 
tlie judgment of the teacher, is not beyond their ability. 

7. Let them compose (make) problems ; also write out 
solutions of problems ; sometimes all in words, without 
figures. 

8. Let them name and describe articles of clothing and 
the materials of which they are made. 

9. Let them write what they know of the school district, 
township, town, county, state, other states, countries, etc. 
These exercises, like the preceding, will serve the double 
purpose of knowledge and language. 

10. Let them tell what they know of some of the most 
prominent or noted men of the country or world. 

11. Let them state what they know of the hisfory of the 
town, township, county, state, country, or anything else 
that has a history. The manufacture of an article from its 
beginning to its end or completion, may be written in the 
form of a history. The manufacture of a pin, needle, knife, 
pen, pencil, thimble, comb, cent, bank-note, hat, book, loaf 
of bread, pie, cake, window-pane, chair, table, sheet of 
paper, piece of cloth, etc., forms suitable subjects for this 
purpose. 

12. Let them describe a picture, telling what they see, or 
imagine they see, in it, and arrange the description in proper 
order, so that any one on reading it or hearing it read 
could form a good mental picture of it. 

13. Let them write letters to imaginary friends, telling 
them what they are doing, have done, or expect to do ; or 
what they are learning about an object, a branch of study, a 
town, county, state, country, man, an event in history, etc. 



General Considerations. 



14. Let them say as many things as they can about an 

object — a dog, cat, cow, horse, pig, sheep, table, chair, stove, 

door, window, clock, looking-glass, sofa, bureau, bench, 

knife, tree, apple, peach, pear, etc.; and lead them to see 

that some of the words are the same in all the lines they 

have written, and that all can be written in one line by 

using the repeated words but once. 

Remark. — These exercises will enable the teacher to intro- 
duce punctuation points and connecting words. 
Examples of 14: 

a. I. The apple is large. 

2. The apple is round. 

3. The apple is red. 

4. The apple is ripe. 

5. The apple is soft. 

Condensed. — The apple is large, round, red, ripe, and soil. 

b. I. The apple is large. 

2. The apple is round. 

3. The apple is ripe. 

4. The apple is not soft. 

Conde7ised. — The apple is large, round, and ripe, but not 
soft. 

c. I. The river is long. 

2. The river is wide. 

3. The river is not deep. 

Condensed. — The river is long and wide, though (or but) 
not deep. 

d. I. The needle is long. 

2. The needle is thick, 

3. The needle is not sharp. 
4. The needle is not hard. 

Condetised — The needle is long and thick, but neither sharp 
nor hard. 

15. Let the pupils say (write) the same thing of as many 
objects as they can, and then write all in one line, or sen- 
tence, with the repeated words used only once. 

Example of 15 : 

1. The cat can run. 

2. The dog can run. 

3. The cow can run. 

4. The horse can run. 

Condensed. — The cat, the dog, the cow, and the liorse can 
run ; or the cat, dog, cow, and horse can run. 



74 Sciefice and Art of Education. 

16. Incorrect expressions, such as are frequently heard 
in conversation, may be interspersed among the language 
exercises for correction — to teach the pupils the correct 
expression. 

17. The following expressions are frequently heard, even 
from the lips of educated persons : 

a. Go and lay down. (Lay down what ?) 

b. He laid on a sofa three weeks. (Laid what on a sofa?) 

c. I lay down every day an hour or two. (Lay down what }) 

d. He set on a log until noon. (Set what on a log?) 

e. He lays in bed too long. (Lays what in bed }) 

f. He comes here most every day. (Most day?) 
g. He thought it was us. (Us was ?) 

h. It was me who said it. (Me was .'*) 

/. The river is raising. (Raising what ?) 

j. He can do it as good as 1 can. (Do it good }) 

k. I know that it was them. (Them was?) 

/. If I was him I would not do it. (When ? Him was?) 
771. It was her that done it. (Her was ? Did she done it ?) 

n. I wish that I was a musician. (When.?) 

o. This is to be divided amongst you and L (Amongst I ? 

p. A person must.be stupid if they cannot do that. (A per- 
son they ?) 

q. It could not have been her. (Her could not have been.?) 

r. He told me that I can go. (How did he know ?) 

s. Is that him ? (Him is .?) 

t. I expect you had a good time last night. (When do you 
expect it?) 

u. I wish that you had went earlier. (Why had went ?) 

V. At what hotel are you stopping at? (Stopping what? 
At stopping at }) 

•w. Who is your letter from? (From who.?) 

X. Who are you writing to ? (To who .?) 

y. Each of the boys have their books. (Each their, and 
each have.) 

z. Leave me alone, I want to sleep. (Why alone you .?) 

18. The various kinds of sentences (according to their 
use) may now gradually be introduced. The statement 
should come first, next the question, then the command, 
and last the exclamation. 

The statement is best taught in comparison with expres- 
sions that are incomplete, that do not assert anything. For 



General Co7isi derations, 75 

example : Some of the rivers of North America are long. 
This is a sentence, because it is a complete statement, a 
complete assertion ; but, some of the rivers of North America^ 
is not a sentence ; it does not state or assert anything ; it 
is simply the beginning of a statement. 

19. The following exercises will furnish a large amount 
of profitable language work : 

a. I. Write a request to your teacher to permit you to 
borrow a book from one of your classmates. 

2. Ask him to permit you to go to the library to consult 
some books of reference ; state the information you desire, 
and in what books you expect to find it. 

3. Ask permission to leave school after you have recited 
your last lesson, giving a full reason for your request. 

4. Ask him to spend the evening at your house, say who 
else has been invited, that you expect a pleasant time, and 
that your parents desire his presence. 

5. Tell him how you spend your evenings ; how much 
time you spend upon each lesson, the order in which you 
study your lessons, your methods of study, what recreations 
you have, and when you retire. 

6. Ask him what the best way is to study each lesson ; 
whether it makes any difference in what order they are pre- 
pared, that is, which comes first, which next, and so on ; 
and say that you will thankfully receive any suggestions 
that he may make concerning your studies. 

b. I. Write to one of your acquaintances, asking what 
school he attends, whether public or private, what branches 
he studies, which he likes best, and how he likes the school. 

2. Ask him what profession or trade he has in view, why 
he has selected it, which of the branches he is pursuing will 
be of most service to him in his life work, and why. 

3. Tell him that you intend to spend next Saturday after- 
noon at fishing, and that you would like to have him ac- 
company you. Tell him what kind of lines and hooks he 



76 Science and Art of Education. 

will need, what kind of bait, where you intend to go, and 
where he shall meet you. 

c. Ask one of your classmates or acquaintances to take a 
ride into the country with you ; name the objects of in- 
terest you expect to see ; speak of the enjoyment his com- 
pany will afford you ; state that your father has given his 
consent for you to go, and that you will have a good horse 
and buggy. 

d. I. You are visiting one of your relatives in the coun- 
try. Write a letter to your mother (if you are a girl) telling 
her of any incidents that came under your notice on your 
way, at the railroad station before starting, in the cars, or 
after you had left the cars. Tell her who met you at the 
station, and how you were received by the family — how 
glad they were to see you. Tell her, too, how you spend 
your time, whether you go out into the fields, and what you 
see when you do so. 

2. Write to your sisters and tell them how you enjoy 
yourself ; tell them at what hours you rise and retire, how 
you spend your time before breakfast, what you do after 
breakfast, how the work of the family is done, how well it 
is systematized, what is assigned to each- as a regular duty, 
and how well each performs his or her part. 

3. Write to your father (if you are a boy) and tell him 
what you saw on the way to the station, how long you had 
to wait for the cars and why, whom you met at the station 
and what you learned from them, whom you saw on the 
cars, and anything of interest that may have occurred or 
come under your notice on the way. 

4. Tell him the number of cattle the people you visit 
keep and the kinds, how many horses they have, the kind 
and number of vehicles, the farming implements and where 
they keep them, how they do their work, what crops they 
raise, the condition of the farm and buildings, how you are 



General Considerations. 77 

enjoying yourself, and what new things you have seen and 
learned. 

e. A salesman is wanted in a city clothing house. Write 
a letter to the proprietor asking for particulars concerning 
the position : i. Whether any other duties would be re- 
quired but that of being in the store during the day ; 2. 
At what hour in the morning you would be required to 
begin work and at what hour in the evening close the store ; 
3. Whether any work would be required in the evening 
after closing ; 4. What salary may be expected. 

Write a reply to your letter as if it came from the pro- 
prietor, giving definite information on every point of inquiry, 
and if the reply is favorable — if the position suits you, write 
to the proprietor offering your services, stating your ex- 
perience as a salesman and your ability to win customers 
and make sales. State, also, that, if acceptable, you are 
willing to be on trial a month or more. It would be of 
service to you, too, to state what educational advantages 
you have had, and that you can bring testimonials as to 
your moral character. 

f. You have been invited to an evening party or enter- 
tainment at the house of a friend, but find it impossible to 
be present. Write a letter expressing your regret, and 
wishing your friend a pleasant time. 

g. Write the autobiography of a pin, ball, shoe, nail, 
thimble ; piece of paper, cloth, glass, bread, etc. 

h. You have lost a valuable watch. Write a notice for 
the paper describing the watch, telling where you suppose 
you lost it, stating what reward you will give the finder on 
delivering it to you, and giving your place of business or 
reside.nce — street and number. 

/. You need an office-boy to run errands. Write an 
advertisement for the paper, stating definitely the kind of 
boy you want, what his duties will be, and compensation 
you will give. 



78 Science and Art of Education. 

j. You have received a valuable dog as a present. Write 
a letter to the person from whom you received it, thanking 
him for it, and telling him that you prize it highly, that you 
are fond of hunting, and this is the kind of dog you have 
for some time desired to get. 

k. Write to a city publishing house for the prices of 
books which you need, and ask for the cheapest way of 
sending them — by mail or by express. 

/. Mr. Hix wants to build. Write to him and tell him 
that you will take the contract at as reasonable a sum as 
any competent builder can do it, and that you will guarantee 
him a good job. Give references of persons for whom you 
have done similar work. 

;;/. Write to a city dry-goods store for goods for a spring 
dress, stating the kind and quality you want, ask the prices, 
and say that if the reply you receive is satisfactory, you 
would like to know how to send the money. 

Write a full, definite reply, and say that if you intrust 
your order to the " house " it shall receive prompt attention 
and be in all respects satisfactory. 

Send the order, enclosing the money, and state how the 
goods shall be sent. After you have received the goods, 
acknowledge the receipt of it, and state how well you are 
pleased with your purchase, and that the house may expect 
further orders from you. 

;/. Samuel Poll owes you three hundred dollars, and you 
intend to purchase some property. Write him a letter stat- 
ing that you contemplate making some investments, and 
that you should be glad if he would let you have the money 
by the first of next April. 

Write his reply. He has been unfortunate during the 
year ; the high waters destroyed his crops, carried away his 
fences, and caused an unexpected outlay of money. Be- 
sides, he has lost one of his best horses. He hopes, there- 
fore, that you will give him more time. 



General Consider atio7is. 79 

Since he cannot pay you at the time you named, he should 
give you an interest-bearing note for the money. Write the 
note. 

You need the money and can get it on the note by in- 
dorsing it. Indorse it in full. 

o. Write a receipt for payment in full to date. 

/. Write an order for six different kinds of goods. Re- 
ceipt it. 

q. Write a note of invitation ; also an acceptance of the 
invitation, stating the pleasure it will afford you to spend 
the afternoon and evening with the family and their friends. 

Also write a note expressing your inability to accept the 
invitation, owing to sickness in your family, and express 
the desire that the family shall pay you a visit. 

r. Write a due-bill, and state how the debt is to be dis- 
charged, whether with money or with goods. If with goods, 
name the kinds and quality. 

s. Write a notice of meeting to be held, naming the place, 
the object, the day, and the hour. 

/. Write an advertisement for the sale of household goods, 
stating the place, the day and hour, the articles to be 
offered for sale, and the conditions upon which they are to 
be sold. 

For an additional number of good topics for Language 
and Composition, see Our Language, published by Leach, 
Shewell & Sanborn, 87 Franklin Street, Boston ; Studies 
in English Composition, published by Allyn & Bacon, 
Boston ; Shaw's English Composition, published by Henry 
Holt & Co., 29 West Twenty-third Street, New York. 

20. The classes of words (parts of speech), of which the 
English language is composed, as has already been stated, 
should be taught in connection with reading and language 
exercises, and may be commenced in the Second Pleader. 

The simplest name should be applied to each class of 
words, a name, as far as possible, to which the children can 



8o Science and Art of Education. 

attach the proper meaning. The noun, for example, may 
be called the name-word ; the verb, the telling-word ; 
the adjective, the kind-word ; the preposition, the relation- 
word ; the adverb, the how, when, and where word (as the 
case may be) ; the conjunction, the connecting-word ; and 
so on. 

21. The classification of sentences according to form 
into simple, complex, and compound, may gradually be in- 
troduced. 

22. The various forms of phrases should be taught in 
connection with simple sentences, and those of clauses with 
complex. At the same time it may also be shown that 
phrases and clauses are substitutes for words, and enable us 
to give not only variety to our expressions, but frequently 
greater clearness. 

23. A knowledge of the different forms of sentences, 

phrases, and clauses is best acquired by practice in writing 

them, substituting one form for another, and pointing out 

and discussing their various elements, relations, and uses. 

Remark. — As soon as the pupils can readily recognize the 
simpler grammatical distinctions, they may with some advan- 
tage begin to use a book on grammar for reference. 

24. Substitute equivalent phrases for the underlined 
words in the following exercises : 

a. He passed my sister's house. 

b. She waited anxiously for the doctor. 

c. I could not hear him for the noise. 

d. The directors meet monthly. 

e. He occupies an influential position. 
/. They had erected a brazen image. 

g. Hence you will see the necessity of it. 

//. American ideas are liberal. 

z. The old oaken bucket had fallen to pieces 
j. The doctor is an intelligent man. 

25. For the phrases in the following sentences substitute 
equivalent clauses : 



General Consider atio7is. 



a. The wool of the sheep affords us clothing. 

b. None in the village suspected him of the deed. 

c. With diligence he must succeed. 

d. I will show you the place of my birth. 

e. No one of my acquaintance was in the room. 
/. I will tell you the reason of my acting so. 

g. Do you know the age of the child } 

26. Change the clauses in the following sentences to 
equivalent phrases : 

a. I see no way in which I cap improve it. 

b. Any person who has good manners will be received. 

c. His house stands where the battle was fought. 

d. All stood with uncovered heads while he read the serv'ce. 

e. When they heard the news they jumped for joy. 

27. For the phrases in the following sentences substitute 
words of equivalent meaning : * 

a. He found it of benefit to use. 

b. She calls upon them every day. 

c. He selected it in place of his brother. 

d. Your conduct was like that of a tyrant. 

e. It would be of no use to try again. 

/. They did it without the authority of law. 

28. Make a simple sentence of each set of the following 
elements : 

a. In America the railways are frequently single lines. The 
railways are formed to carry a limited commerce. Sidings are 
provided at convenient situations. 

b. The river overflowed. The river was the Ohio. This was 
in January. It happened on the tenth of the month. 

c. Wodin was the chief god of the Old English people. 
Wodin was by the Danes called Odin. He was the chief giver 
of valor. He was the chief giver of victory. 

d. John Wycliffe did a great work. This great work was a 
translation of the Bible. The translation was made by himself. 
He was assisted by several friends and followers. 

e. The English fearlessly boarded the ships. The ships were 
those of the enemy. They cut the rigging. They gained the 
victory. The victory was easily won. 

/. The next morning the battle began in terrible earnest. 
The next morning was the 24th of June. The battle began at 
break of day. 

g. Columbus returned to Spain in 1493. He had spent some 



82 Science and Art of Education. 

months in exploring the delightful regions. These regions bad 
long been dreamed of by many. These regions were now first 
thrown open to European eyes. Columbus had been absent 
seven months and eleven days. 

29. Combine each of the following sets of simple sen- 
tences into a complex sentence. 

a. Tin is a metal. Ancient Britain was most famous for tin. 
The Phoenicians were first induced to visit Britain for tin. 

b. He spoke to the king like a rough man. I think this my- 
self. He was a rough angry man. He did nothing more. 

c. The ingenuity of man has- made a lever of the mind. This 
lever spares him an immensity of toil. This lever is applied to 
machinery. 

d. The ships of Sesostris, the Egyptian conqueror, were 
formed of cedar. One of these ships was 280 cubits long. 
Ancient writers notice this. The gigantic statue of Diana in 
the temple of Ephesus was also formed of this timber. 

e. Andrew Douglas was willing to share the danger and the 
honor. Andrew Douglas was master of the Phoenix. He had 
on board a great quantity of meal from Scotland. 

f. Three or four bounds of the horse carried us out of reach 
of the enemy. The enemy quickly resumed his flight. The 
enemy had merely turned in desperate self-defence, 

g. God in his good has covered the earth with herbs and 
trees. We inhabit the earth. These herbs and trees furnish 
us with food, clothing, and other articles. These articles con- 
tribute to our comfort and luxury. 

h. At length the mystery of the ocean was revealed. The 
theory of the great navigator was triumphantly established. 
The theory had been the scoff of sages. He had secured to 
himself glory, This glory must be durable. The world itself 
is durable. 

30. Change the following simple sentences to compound 
sentences : 

a. The Rhone, flowing into the Lake of Geneva, emerged 
from it at the town of the same name. 

b. These events, trifling doubtless in the estimation of the 
reader, were affecting to me in the highest degree. 

c. Snatching the handkerchief, he quickly wrapped it around 
the wounded part. 

d. The trees met overhead, forming an archway. 

e. On further consideration I have decided to remain. 

/. After a moment's reflection he proceeded to pass sentence. 
g. The king, a man of rare vigor, would allow no foreigr? 
prince to encroach on his rights. 



General Considerations, '^^i 

h. In forwarding your own interests, do not impede those of 
others. 

/, The coral insect, barely possessing life, is hourly creating 
habitations for men. 

31. Combine each set of the following simple sentences 
into a compound sentence : 

a. They next erected a crucifix. They prostrated themselves 
before it. They returned thanks to God. God had conducted 
their voyage to such a happy issue. 

b. He possessed quick perceptions. He observed accurate- 
ly. He was able to place his hand on the right animals. He 
did so without hesitation. 

c. The island at first seemed uninhabited. The natives grad- 
ually assembled in groups upon the shore. The natives grad- 
ually overcame their natural shyness. The natives received us 
very hospitably. They brought down for our use the various 
products of their island. 

d. The struggle was now at an end. The inhabitants were 
terror-stricken. They burst through the lines. They fled in 
every direction. 

e. They saw their leader fall. They thought him slain. 
They at once gave up the contest. This was in accordance 
with the practice of their ancestors. 

f. Steam has increased indefinitely the mass of human com- 
forts. Steam has increased indefinitely the mass of human en- 
joyments. Steam has rendered cheap the materials of wealth 
and prosperity. Steam has rendered accessible the materials 
of wealth and prosperity. It has done so all over the world. 

g. The sun then broke out. The sun dispersed the vapor 
and the cold with his welcome beams. The traveller felt the 
general warmth. The sun shone brighter and brighter. The 
traveller sat down. The traveller was overpowered by the 
heat. The traveller cast his cloak upon the ground. 

32. Change the following compound sentences to com- 
plex sentences: 

a. You have asked me a question and I have answered it. 

b. The statement is false and he knows it. 

c. They did not know their lesson and so he kept them in. 

d. Finish this and then I will let you go. 

e. He was very ill, but still he tried to finish it. 
/. A boy had seen it fall and had picked it up. 

g. He tried several keys, but none of them would fit it. 

For a greater variety of exercises in the various kinds of 
sentences, see First Steps and Second Steps in English 



84 Science and Art of Education. 

Composition, published by W. Stewart & Co., London, 
England ; Practical Exercises in Composition, and Exer- 
cises in English, by H. I. Strang, the former published by 
The Educational Publishing Co., Boston, the latter by D. 
C. Heath & Co., Boston. Kerl's Composition and Rhet- 
oric is also a good book for sentence and composition 
work ; published by American Book Co., New York. 

33. Exercises in the discrimination of words should 

form part of the sentence work. 

Write sentences in which the following words shall be cor- 
rectly used : Raise, rise ; sit, set ; bring, fetch ; fly, flee ; flow, 
flew, fled ; shut, close ; board, feed ; hung, hanged ; lay, lie ; 
leave, let; lend, borrow; lose, loose ; teach, learn; wring, ring; 
begin, commence; forsake, desert; load, burden; empty, va- 
cant ; sleigh, slay ; expect, suspect, suppose ; believe, calculate ; 
may, can ; fix, repair, mend; think, guess; enjoy, possess ; be- 
tween, among; invent, discover; handsful, handfuls ; preacher, 
minister, pastor, clergyman ; sow, sew ; luck, success ; station- 
ary, stationery ; desert, dessert; continual, continued; compo- 
sition, essay ; there, their; hard, difficult; balance, remainder ; 
advice, advise ; all, awl ; aloud, allowed ; altar, alter ; ant, aunt; 
ascent, assent; assistance, assistants; bail, bale; bait, bate; 
bald, bawld ; bear, bare ; base, bass ; beat, beet ; blew, blue ; 
bawl, ball; burrow, borough; bough, bow; cannon, canon; 
capital, capitol ; ceiling, sealing; cell, sell; cent, sent, scent; 
chord, cord; site, cite, sight ; climb, clime; council, counsel; 
fare, fair; gait, gate; great, grate; holy, wholly; gage, gauge; 
hail, hale; pale, pail; pane, pain; plane, plain; pray, prey; 
rain, rein, reign; sale, sail; strait, straight; vane, vain, vein ; 
pare, pear, pair ; canvass, canvas ; tacks, tax ; claws, clause ; 
nought, naught ; feet, feat, leave, lieve ; meet, mete, meat ; 
peel, peal ; pleas, please ; seed, cede ; seas, sees, seize ; heard, 
herd; lesson, lessen ; miner, minor; petition, partition ; of, oft". 

34. The properties that characterize well-written sen- 
tences, paragraphs, and essays, under the usual titles of 
purity, propriety, precision, and clearness, strength, unity, 
and harmony, should not, as is usual, be postponed until a 
late period of a pupil's school life, but should as early as is 
possible be introduced by directing his attention to merito- 
rious as well as to faulty constructions, and demanding cor- 



General Considerations. 85 



rections or improvements so far as he is capable of seeing 
their force and making them. 

35. The only sure way of training pupils to the careful 
use of English is to begin as early as their years permit, and 
to demand that every exercise of theirs shall be as nearly 
perfect as they can make it ; and this course must be con- 
tinued to the end of their school days. 

36. Essays* — When pupils have acquired sufficient power 
of thought and skill in expression to write essays, subjects 
for the purpose may be assigned them. Care must, however, 
be taken that no abstract subjects, no subjects beyond their 
comprehension, nor anything in which they cannot be inter- 
ested, be given. 

37. After a general subject has been selected or assigned, 

it should be considered in all its bearings, and some special 

line of thought or view of it decided upon. 

Remark. — Writing upon general subjects cannot end in any- 
thing definite. 

38. Having determined upon the line of discussion, the 
next thing is to keep it before the mind until it has been 
thoroughly thought over and everything found that has a 
direct bearing upon it. This done, a careful, logical out- 
line of the main and sub-topics should be made, containing 
nothing not strictly in accord with the determined line of 
thought. 

Remark. — Finding the matter for the essay and making the 
outline constitute the most important and difficult part of the 
work. 

39. Next comes the writing, the composition. A good 
plan to pursue is, to write a little essay, as it were, upon 
each main topic, combine them into one, look it over to 
correct errors, then lay it away for a few days or a week 
before re-examitiation and rewriting. An essay should be 
several times carefully examined and rewritten before it is 
handed to the teacher for inspection and suggestions. 



86 Science and Art of Educatio7i. 

40. Unless intended for a public audience, no essay 
handed to the teacher should be corrected by him ; only the 
place where an error exists should be indicated by some 
general mark or sign, and the pupil, at least at first, left to 
discover the fault himself. It is only by practice in discov- 
ering that discovering is learned. 

41. An essay should be handed two, three, or more times 
to the teacher for inspection and criticism, and as many 
times rewritten. In short, it should be criticised and re- 
written until it is free from errors. 

In few things, if in any, do we find more failures in 
teaching than in composition, and the best book on essay 
writing is a competent teacher. 

The following books on composition, in addition to those 
already named, may be used with advantage by teachers : 
Longmans' School Composition, published by Longmans, 
Green & Co., New York ; The Foundations of Rhetoric, 
published by Harper & Brothers, New York ; Composition 
and Practical English, by William Williams, published by 
D. C. Heath & Co., Boston. For lower grades of schools, 
the following will prove serviceable : Stories for Composi- 
tion, published by Educational Publishing Co., Boston ; 
How to Write a Composition, published by Dick & Fitzger- 
ald, New York ; Primary Reproduction Stories, and Hall's 
Composition Outlines, published by A. Flanagan & Co., 
Chicago ; The Writing of Compositions, published by E. 
L. Kellogg & Co., New York ; and W. B. Powell's whole 
series of language books, published by E. H. Butler & Co., 
Philadelphia. 

Nearly all the foregoing books can be had of E. L. Kel- 
logg & Co., 6i East Ninth Street, New York. 



SUGGESTIONS FOR TEACHING 
NUMBERS. 

Teach the concept (idea) concretely, with pebbles, beans, 
grains of corn, shoe-pegs (colored or plain), spools, squares, 
cubes, balls (spheres), cylinders, triangles ; in short, with 
any suitable objects that may be had. A variety of objects 
should be kept on hand for this purpose. 

Figures should not be introduced until the children can 
work well with objects, pictures, etc., and until they will 
not confound figures with numbers. 

Color and form should be taught in connection with 
numbers. 

Lesson on Two. 
a. What the Pupils must Discover. 

1. 1 + 1= 5. 2— 2= 9. fofl = 

2. I X 2 = 6. 2 ^ I = 10. i of 2 = 

3. 2 X I = 7. 2 -=- 2 = II. f of 2 = 

4. 2 — I = 8. i of I = 12. f = 

b. For seat work the following notation may be used with 
children that are not far enough advanced to write words 
or to use figures : 

1. i + i =11 5. 11 — 11=0 9. 1(1 ) =f = I 

2. iXii = ii 6. ii-M = II 10. i (11) =i + i=i 

3. iiXi =11 7. 11-^11= I II. f (11) =1 + 1=11 

4. II — I =1 8. i (i) =i 12. f =1 

Note. — i. The teacher should substitute some form of the 
concrete for the fractions in the seat-work, until the children 
can use figures. A short vertical line divided into two equal 

87 



Science and Art of Education. 



parts, with a nought covering the upper part, might represent 
\; both parts uncovered,!; the line divided into three equal 
parts, with the upper part covered, |; with the two upper parts 
covered, \, etc, 

2. Horizontal lines, squares, triangles, circles, and pictures of 
suitable objects may be used by the children in performing the 
fractional work ; the whole object or picture representing or 
being the unit, and the parts the fractions. 

3. In solving problems in which a fractional part is required 
of a number consisting of several ones, or units, the children 
should first be taught to find the sum of the fractional parts 
of the separate units; later, after they can readily do this, 
they should be taught the usual way of obtaining the result. 
Problem 10 of the foregoing seat-work may be solved with two 
horizontal lines, one of the halves of each being covered with a 
nought or some other device to indicate that it is not to be 
counted. It may also be solved with squares, as follows : 

c. Suggestive Questions and Problems. 

1. Give me a cube. Give me another. How many have 
I now? (One and one are called two; or, simply, are two.) 

2. Show me two fingers, two hands, two boys, a two of 

red cubes, a two of pencils, a two of anything else. 

Remark. — A two of anything means two things taken to- 
gether and considered as a unit. 

3. One bird and one bird are how many birds ? 

4. Show me a two. Show me a one. How many ones 
are in a two ? 

5. If two birds are on a tree and one of them flies away, 
how many remain ? 

6. What number is one more than one ? 

7. What number is one less than two ? 

Remark. — Every operation should be proved or illustrated 
with objects or drawings of them. 

8. How many twos are in two ones ? How many ones 
does it take to make a two ? 

9. From a two take away two ones, and what have you 
left_? Prove it with yellow cubes. 



Suggestions for Teaching Numbers. S9 

10. From two ones take away a two, and what have you 
left ? Prove it with blue cylinders. 

d. Suggestive Dialogue to Teach the Half. 

Teacher. If you wanted to give me half of an apple, how 
would you cut (or divide) the apple ? 

Pupil. I would cut it through the middle. 

T. Which one of us would get the larger piece ? 

P. Neither; one piece would be as large as the other. 

T. Make a picture-apple upon the blackboard, and draw 
a line through it where you would cut it. 

T. Here are two pieces of cardboard; which of them is 
the longer ? 

P. One is as long as the other. 

T. How much longer are the two pieces together than 
this piece ? 

P. They are just as long. 

T. If I should cut the long piece across the middle, which 
of the two parts would be the longer ? 

P. Neither piece would be longer than the other. 

T. What part of the whole piece would each of the 
parts be ? 

P. One half. 

T. What did I do ? 

P. You cut the long piece through the middle. 

T. How can you tell that I cut it through the middle ? 

P. By trying whether one piece is as long as the other. 

T. You may try it. 

P. One piece is just as long as the other. 

T. What part of the whole piece do I hold in my hand ? 

P. One half. 

T. One half of what ? 

P. Of the whole piece. 

T. Lay one piece at the end of the other, and see what 
the two halves make. 



9© Science and Art of Education. 

P. They make the whole piece. 

T. Since these two pieces make the whole piece, one 
cji them is what part of two ? 

P. One half. 

T. How do you know that each piece is one half of the 
two pieces ? 

P. Because it is one half of the whole piece, and the 
whole is made of the two pieces. 

T. Which of these two yellow cubes is the larger ? 

P. One is as large as the other. 

T. How do you know ? 

P. I have tried them. 

T. Now since one of the two pieces of cardboard is one 
half of the two pieces, what part do you think one cube is 
of the two cubes ? 

P. One half. 

T. One apple is what part of two apples ? 

P. If they are of the same size, it is one half of them. 

T. How many halves are in the whole of anything ? 

P. Two. 

T. How can I get one half of anything — of an orange, 
for example ? 

P. By cutting it through the middle. 

T. Why cut it through the middle ? 

P. To make one piece as large as the other. 

T. Why must one piece be as large as the other ? 

P. If they were not of the same size they would not be 
halves. 

T. How can I get one half of two candies ? 

P. By breaking each one into two equal pieces, and tak- 
ing a piece of each of them. 

T. Cculd you get a half of the two in any other way ? 

P. If one of the candies is as large as the other, one of 
them would be a half of the two. 

T. Which one of them ? 



Suggestions for Teaching Numbers. 9I 

P. Either one. 

T. How can you tell ? 

P. Because one is as large as the other. 

T. Are all halves of the same size ? 

P. Yes, they are. 

T. You may draw two picture-apples upon the black- 
board, making one larger than the other. 

T. Draw a line through the middle of each, and then tell 
me whether the halves of the small one are as large as those 
of the large one. 

P. No; they are not. 

T. Do you still think that all halves are of the same size ? 

P. No; I do not. 

T. What halves, then, are of the same size ? 

P. Those of the same thing, or of things of the same size. 

T. In how many ways could you cut this card (parallel- 
ogram, I in. by 2 in.) into halves ? 

T. I will give you a card and you may draw a pencil- 
mark across where you would cut it, and then tell me in 
how many ways you could cut it ? 

P. I could cut it in two ways, lengthwise and crosswise. 

T. I will give you another card, and you may cut one of 
them lengthwise and the other crosswise, and then tell me 
whether all the halves are of the same size ? 

P. No, they are not ; the pieces of the card cut length- 
wise are longer than those of the one cut crosswise. 

T. Do you notice any other difference ? 

P. Yes, the short pieces are broader than the others 

T. Find out how much broader they are ? 

P. They are twice as broad. 

T. Now cut one of the short pieces into two equal pieces 
and see whether the two parts laid together make a piece 
as long as one of the long pieces. 

P. Yes, they do. 

T. What can you again say of the halves of anything ? 



9^ Science and Art of Education, 

P. That one is as large as the other. 

T. In how many ways did you cut your card into halves ? 

P. In two ways. 

T. What are they ? 

P. Lengthwise and crosswise. 

T. Can you cut them in any other way so that the two 
parts will be of the same size ? 

P. I think I can, but I am not sure of it. 

T. You may try it. 

P. Yes, I can cut them in another way — I can cut them 
from one corner across the middle to the opposite corner. 

T. How do you know that the pieces are of the same 
size ? 

P. I laid one upon the other and it covered it exactly. 

T. Show me half of your cubes. 

T. Of how many do you show me a half ? 

P. Of two. 

T. How many halves does it take to make the whole of 
anything ? 

P. Two. 

T. Two what ? 

P. Two halves. 

T. To make what ? 

P. To make the whole of anything. 

e. Suggestive Dialogue to Teach the Applications in Two, 

T. What do I hold in my hand ? 
P. A two-cent piece. 

T. How many cents would you give me for it ? 
P. Two. 

T. How many cent-candies could you get for one cent ? 
P. One. 

T. Illustrate (show) it with pictures upon the black- 
Doard ; also with toy-money and crayons. 



Suggestions for Teaching Nutnbers. 93 

T. How many cent-apples could you get for tv, o cents ? 

P. Two. 

T. Prove (show, illustrate) it with picture-apples and 
picture-cents. 

T. If I should send you to the store to buy two slate- 
pencils that cost a cent each, how much money would you 
pay for them ? 

P. Two cents. 

T. Your sister sends you to the store with a two-cent 
piece to buy an orange that costs a cent; how much money 
will you bring back ? 

P. One cent. 

T. If you should go to the post-office with two cents to 
buy cent-stamps, how many would you get? 

P. Two. 

T. Henry has two rabbits, and this is one more than 
Sarah has ; How many has Sarah ? 

P. One. 

T. John has half as many roses as his sister ; if he has 
one, how many has she ? 

P. Two. 

T. I know a number whose double is two, what is it ? 

P. One. 

T. What number is that whose half is one ? 

P. Two. 

Lesson on Three. 
a. What the Pupils must Discover. 



1. 


2 + 1 = 


^' z-^z^ • 


15- fof 2 = 


2. 


1 + 2 = 


9. 3 -^ I = 


16. 1 of 2 = 


3- 


1X3 = 


10. 3-^2 = 


17. iof 3 = 


4- 


3X1 = 


II. ^ of I — 


18. f of 3:::. 


5- 


3 - I = 


12. 1 of I =: 


19. i of 3 = 


6. 


'3- 2 = 


13. 1 of I = 


20. t of 3 =: 


7- 


z- Z'^ 


14. ^ of 2 =; 


21. 1 Of 3 == 



94 Science and Art of Education. 

Remark. — I. In 3 -4- 2, the question is, how many twos are 
in three ; and the answer is, one two and one one, and may be 
written thus, i (i), the parenthetic part being the remainder. 

2. The yard with its parts in feet should be taught in connec- 
tion with the foregoing. 

b. For seat-work the following will serve as example : 

1. ii-fi =111 4. Ill xi = iii 7. 111 — 111=0 

2. I +ii = iii 5. Ill — 1= II 8. iii-f-iii = i 

3. IX 111 = 111 6. Ill — ii = i 9. iii-i-i = iii 
lo.iii -^ ii = i(i) 16. i(ii) =1+1=11 

11. \ (I) =i 17. i(iii) = i+Ki=l=ii 

12. Id) =1 18. |(iii) = |+|+|=iii 

13. f (0 =1=1 19. i(iii)=i+i+i=l= I 

14. \ (ii)=i +i=f 20. |(iii)=^ + i+|=ii 

15. |(ll)=|+|=li 21. 3(111)^3 + 3+1^11, 

Note.— The method of solution is indicated in all the exer- 
cises that follow the tenth. 

Remark. — i. As before remarked, instead of the foregoing 
notation, horizontal lines, squares, triangles, circles, and pict- 
ures may be used for performing all the fractional work. For 
examples, the 20th of the foregoing may be solved thus, with 
squares : 

tOt!I]+ID=ID=D]] n D 

Remark. — 2. The pupils should be required to make stories 
of their exercises. Of No. i of the foregoing the following 
may be made : I had two cents and my sister gave me another; 
then I had three. Or, Sarah had two roses and her mother 
gave her another ; how many had she then } 

c. Suggestive Questions and Problems. 

1. Give me two red spheres (balls). Give me another. 
How many have you given me altogether ? (Two and one 
are three). 

2. Show me three fingers, three cylinders, three girls, 
three boys. 

3. What number is one more than two ? 

4. Three is one more than what number ? 



Suggestions for Teaching Numbers, 95 

5. One is two less than what number ? 

6. Wliat number is less than three ? 

7. Wiiat number is two more than one ? 

8. To what number must I put (add) one to make 
tliree ? 

9. One and one and one are how many ones ? 

10. Show me a two, also two ones ; which is the larger? 

1 1. How many ones are in a two ? In a three ? 

12. How many twos are in a three, or in three ones? 

13. Make three in all the ways you can. 

14. Make three picture-boys on your tablet, three pict- 
ure-girls, three picture-horses, three picture-wagons, three 
picture-cats. 

15. Under each of three trees John found an apple; how 
many apples did he find ? 

16. Three mice were in a box and Charles killed all but 
two ; how many did he kill ? 

17. Henry had three roses and gave all but one to Alice; 
how many did Alice receive ?' 

18. Frank has three horses and one saddle, how many 
more horses has he than saddles ? 

19. Ella had three playmates and gave to each a pear; 
how many pears did she give away? 

20. Jacob caught three rabbits in three traps ; how many 
did he catch in each ? 

21. If you should lay three grains of corn upon the floor 
and a mouse should come and carry one of them away at a 
time, how many trips would it have to make to carry all of 
them away ? 

22. Place three grains of corn upon the floor ; now, if a 
mouse could carry away only one grain at a time, how many 
mice would be required to carry all of them away at once ? 
Make picture-mice and prove it. 

23. Place three buttons upon the table ; take them away 
two at a time ; how many times did you take two away ] 



96 Science and Art of Education. 

24. How many twos in three? How many ones? How 
many threes ? 

25. Make two in all the ways you can. 

26. Make ihree of ones ; how many does it take ? 

27. How many ones in two? In three? How many 
twos in two ? 

28. How many ones in one half of two ? In three halves 
of two ? 

• 29. How many twos in two ones ? In three ones ? 

30. Take three buttons and lay each one by itself ; how 
many separate buttons have you laid ? Count them. Each 
one of the three is what part of all of them ? (It is one of 
the three equal parts, or one third.) 

31. Show me one third of three yellow squares; two 
thirds of three buttons ; three thirds of three picture- 
apples. 

32. How can you get one third of anything ? Can you 
get one third of one ? How ? Do it. 

33. How would you get one half of anything ? 

34. Which is the larger (or greater), one , C^ ^ 
half or one third ? Prove it. V_ii-^ 

35. How many halves in one and one half ? 

36. If I cut an apple into two equal pieces, what is one 
of the pieces called ? What is each of the pieces called ? 

37. Show me how you would get two thirds of anything. 
Two is what part of three ? One is what part of two ? 

38. This stick (3 inches) is how many times as long as 
that (i inch) ? How can you find out ? Do it. 

39. This stick (i inch) is what part of that (3 inches). 
How can you tell ? 

40. This stick (i inch) is what part of that (2 inches.) ? 

d. Suggestive Practical Business Problems. 

We will now play store. John may be the merchant and 
the others his customers or buyers, 



Suggestions for Teaching Numbers. 97 

1. Sarah may buy two buttons, at a cent apiece, and give 
him a three-cent piece for them. How much change will 
he give her ? 

Remark. — The more these transactions are made like real 
business the more interest the children will take in them. 

2. Alice may buy a one-cent candy and a two-cent candy 
How many cents must she give for them ? 

3. Henry buys three pencils at a cent each ; how much 
money will pay for them ? 

4. Fannie wants three cents' worth of eggs, and the eggs 
cost a cent each ; how many will she get ? How many 
would she get for two cents ? Illustrate. ^ ^ 

5. How many two- cent rings can I get for three cents ? 

6. How many nuts at a half-cent each could you get for 
one cent ? How many for a cent and a half? 

7. Fill this pint measure with sand and pour it into that 
quart measure. Did it fill it ? Fill it again and pour it in. 
Is it now full ? What did I call this measure ? What that ? 
How many times this did it take to fill that ? How many 
pints in a quart ? In a half-quart ? 

8. One pint is what part of a quart ? Two pints are what 
part of a quart ? We will call this water milk, and Annie 
may sell it at a cent a pint. 

9. Alice wants a quart and gives Annie a three-cent 
piece ; how much change will she receive ? 

10. Sarah wants two pints ; how much will they cost ? 

11. Jacob wants a half-quart and has a two-cent piece 
with which to pay for it ; give him the change. 

12. John may take this foot-measure and see how many 
such could be made of that yard-measure. 

13. Henry, you may tell us how many foot-measures it 
takes to make a yard-measure, or a yard. Prove it. 

14. How many feet make a yard, or are in a yard ? How 
many in one third of a yard ? In two thirds ? In one half ? 
Sarah may now sell tape to the other members of the class. 



98 Science and Art of Education. 

Remarks. — Let the children make tape of newspapers, by 
cutting them into narrow strips and sewing the ends together. 

15. Alice buys a yard that costs two cents and a yard 
that costs one cent ; how much will it all cost ? 

16. Anna wants a third of a yard of that which costs 
three cents a yard ; how many cents must she give for it ? 
Prove it ? 

17. Fannie wants a cent's worth of that which costs three 
cents a yard ? How much will she get ? 

18. Peter wants a yard and a half of that which sells at 
two cents a yard ; how much money will pay for it ? 

19. Ella had three yards and sold one third of them ; 
how many had she left ? If it sold at three cents a yard, 
what did she get for what she sold ? 

20. When a quart of milk costs two cents, what does a 
pint cost ? 

21. How many pints in a quart ? How many quarts in 
three pints ? 

22. How many feet in a yard ? In a half -yard ? In a 
third of a yard ? 

23. One foot is what part of a yard ? 

24. If a yard of tape costs three cents, what does one 
foot of it cost ? What two thirds of a yard ? 

Lesson on Four. 
a. What the Puils should Discover. 



I. 3 + 1 = 


9-4-4= 


17. |of4-= 


25. I qt. 


2. 1+3= 


10. 4-^2== 


18. iof 1 = 


26. \ gal. 


3; 2 + 2 = 


II. 4-5-1 = 


19. fof 1 = 


27. 4 gills 


4. 2x2= 


12. 4-^3= 


20. f of I = 


28. i pt. 


5. 1x4= 


13, 4-^4= 


21, fof 1 = 


29. 4 pk. 


6. 4—1 = 


14. iof4= 


22. \ of i= 


30. i bu. 


7. 4-2= 


15, iof4= 


23, 4qts.= 


31. I pk. 


8. 4-3=^ 


16, |of4= 


24. 3 qts. =; 


32. 4 weeks 



Suggestions for Teaching Numbers. 99 

b. The following will serve as examples of what may be 
given as seat-work : 



I. 


III 


+ 1 = 


10. 


IIII-MI = 


18. 


\ (I ) = 


o 


I 


+ 111 = 


II. 


IIII-^I = 


19. 


f (I ) = 


3- 


II 


+ 11 = 


12. 


Illl-f-III = 


20. 


f (I ) = 


4- 


II 


X n = 


13- 


IIII-4-IIII = 


21. 


t (1 ) = 


5- 


I 


X 1111 = 


14. 


^ (inn = 


22. 


i(J.iii) = 


6. 


nil 


— I = 


15. 


i (nil) ^ 


23. 


f(iiii)= 


7. 


nil 


— II = 


16. 


f (nil) = 


24. 


f (iiii) = 


8. 


nil 


— III = 


17. 


1 (nil) = 


25. 


i(i ) = 


9- 


nil 


— 1111 = 











Note. — In No. 25, the question is, What part of one is one 
half of one half of one ? and by dividing each of the two halves of 

a given line or square into two equal parts, thus, |^, \ \ \ 
the answer is found to be one fourth. 







)k 





Remark.— The pupils should also be required to write the 
foregoing exercises in words, either in the question or in the 
statement form or in both. This work will give them practice 
in penmanship, spelling, and language, in addition to that of 
numbers. Nos. i, 10, 14, 15, and 24, for example, maybe written 
as follows: (i.) Three and one are how many.^ or three and 
one are four. (10.) How many twos are in four.'* (14.) One 
half of four ones (or of four) is two ones (or two). (15.) What 
is one third of four ones? (24.) What are three fourths of four 
ones, or of four (ones being understood) } 

c. Suggestive Questions and Problems, 

1. Take a one and a two. How many ones have you 
taken ? How many threes ? 

2. Take a three and a one. How many ones did you 
take ? (Three and one are four.) 

3. Give me a two. Give me another. How many ones 
did you give me ? How many ones are in four ? How 
many twos ? How many threes ? How many fours ? 

4. Show me one four, one three, one twp, 

5. How many twos in three ? 



Science and Art of Education, 



6. One half is what part of one and one-half ? Of two 
ones ? 

7. How many ones in three halves ? In four halves ? 

8. How many feet in a yard ? In a half-yard ? In two 
thirds of a yard ? 

9. How many pints in a quart ? In a half-quart ? 

10. How many quarts will fill this gallon measure ? How 
can you find out ? Do it. We will call this water milk, and 
sell it at four cents a gallon. 

11. Sarah wants a quart and gives a two-cent piece to pay 
for it ? how much change shall she receive ? 

12. Alice wants a half-gallon ; how many cents will pay 
for it ? 

13. Jacob buys two pints ; what will they cost ? 

14. Anna wants two half-gallons ; how much money will 
she need to pay for them ? How many two-cent pieces will 
pay for them ? 

15. Here is a measure we have not yet used; it is called a 
peck, and is used for measuring apples, peaches, nuts, corn, 
wheat, etc., things not liquid and not sold by the pint or 
quart. Henry may fill it with saw-dust and empty it into 
that half-bushel. Does it fill it ? Peter may fill it and also 
empty it into the half-bushel. Is the half-bushel now full ? 
How many pecks make (fill) a half-bushel ? How many a 
bushel (whole bushel) ? 

We will now sell peaches, at a dollar a bushel You may 
get the money (toy-money). John may sell them. He has 
no peaches, but he may make believe (pretend) that he is 
selling and you may make believe that you are buying. 

16. Jacob wants two pecks. He may show us the 
money he will give for them. How many quarter-dollars 
has he ? Has he enough money ? Prove it. How many 
half-dollars would pay for them ? Prove it. 

17. Peter sells pears, at two dollars a bushel, and John 



Suggestions for Teaching Numbers. loi 

buys a peck. He may get the money to pay for them. Did 
he get enough ? 

1 8. Jacob wants a bushel and a peck of pears. He may 
get the money to pay for them. Has he enough money ? 
How can you tell ? 

19. Three quarters of a bushel make how many half- 
])ushels ? How many pecks ? 

20. How many half-dollars will pay for three quarters of 
a bushel of pears ? What part of a dollar would pay for 
them ? 

21. Alice wants a bushel and a half ; how much money 
will pay for them ? Prove that your answer is right. 

, 22. Sarah sells Henry two and a half yards of cloth, at 
one dollar a yard ; how many dollars will pay for them ? 
How many dollars will three half-yards cost ? how many 
three quarters of a yard ? 

23. What number diminished by two leaves nothing ? 

24. From what number can I take one-half and have 
one and one fourth left ? 

25. From what number can I take two and one half and 
have one and one half left ? 

26. What number added to one makes four ? 

27. What number added to one and one half makes 
three ? 

28. I think of a number whose half is two ; what is it? 

29. What number taken four times makes four ? 

30. I think of a number whose fourth part is one ; what 
is it? 

31. What two numbers make four? What three num- 
bers ? 

32. What number taken three times, and ^one added, 
makes four? 

33. What number doubled makes four? What three ? 

34. What number doubled, and one added, makes three ? 

35. Make four in all the ways you can, mentally. 



I02 



Science and Art of Education* 



36. Make two in all the ways you can ; also three. 

37. What is one half of one? of three? 

38. What is one third of one? of two ? of four? 

39. What are two thirds of one ? of two ? of four ? 
of three ? 

40. What are two fourths of one ? Three fourths of 

one ? Three fourths of two ? of three ? of four ? 

Remark. — Whenever it is possible stories should be made 
of abstract problems. Stories give reality and interest to the 
work. 



Lesson on Five. 



a. I. 4 4- I = 


15. 5-^ 2 = 


29. f of 2 


2. I + 4 = 


16. 5 -^ 3 = 


30. i of 3 


3- 3 + 2 ^ 


17. 5 -^ 4 = 


31. fof 3 


4- 2 + 3 = 


18. 5 - 5 = 


32. iof 3 


5. 2 + 2 + I = 


19. i of 5 = 


ZZ- i of 4 


6. 2X2 + 1 = 


20. i of 5 = 


34- 1 of 4 


7. I X 5 = 


21. 1 of 5 = 


35- 1 of 4 


8. 5 X I = 


22. :J of 5 = 


36. f of 4 


9- 5 - I == 


23. f of 5 = 


37- \ of 5 


10. 5 — 2 = 


24. -J^ of I = 


38. 1 of 5 


II. 5-3 = 


25. 1 of I = 


39- 1 of 5 


12. 5-4 = 


26. 1 of I = 


40. i of 5 


13- 5 - 5 = 


27. i of 2 = 


41. 1 of 5 


14. 5-^1 = 


28. 1 of 2 = 





b. Examples of Seat-ivork. 

Remark. — For the seat-work of five and the following num- 
bers, until the children can use figures, the notation of the 
numbers preceding five, or the following, may be used. 



1. four -f one = 

2. one 4- four = 

3. three + two = 

4. two + three = 

5. two + two + one = 

6. two X two + one = 

7. one + two X two = 

8. one X five = 

9. five — one = 

10. five — two = 

11. five — three = 



12. five — four = 

13. five — five = 

14. five -f- one = 

15. five -^- two = 

16. five -^ three = 

17. five -4- four = 

18. five -f- five = 

19. one half of five = 

20. one third of five = 

21. two thirds of five = 

22. one fourth of five = 



Suggestions for Teaching Numbers, 163 

23. three fourths of five = 33. one fifth of four = 

24. one fifth of one = 34. two fifths of four = 

25. three fifths of one = 35. three fifths of four = 

26. five fifths of one = 36. four fifths of four = 

27. one fifth of two = 37. one fifth of five = 

28. three fifths of two = 38. two fifths of five = 

29. four fifths of two = 39. three fifths of five = 

30. one fitfh of three = 40. four fifths of five = 

31. two fifths of three = 41. five fifths of five = 

32. four fifths of three = 

c. Suggestive Questions and Froblefns. 

1. Place four blocks upon the table. Place another on. 
How many are on now ? (You have placed five on.) How 
many did you place on first ? How many afterwards ? 
How many altogether? Then four and one are how 
many? 

2. Can you make five of ones ? Try it. How many 
does it take ? 

3. Can you make five of twos ? How many does it take ? 
Then two twos and one are how many ? How many ones ? 

4. Can you make five of threes ? How make does it 
take ? One three and one two are how many ? One three 
and two ones are how many ? 

5. Can you make five of fours ? How many does it 
take? 

6. Make five in all the ways you can (in your mind) 
without objects, and in every case tell the result. 

7. How many threes are in four ? How many twos ? 
How many ones ? 

8. What is one half of four ? of two ? of one ? One 
third of three ? of four ? of one ? 

9. What is one fourth of four ? of three ? of two .' of one ? 
One third of two ? 

10. Divide five pebbles into five equal parts. How many 
have you in each ? One is what part of five ? How many 
such parts are in five ? How many fifths of five are in 
five ? How many fifths of one are in one ? 



104 



Science and Aft of Education. 



II. How many times can you find two fifths of one in 

one? 

Remark. — Problems of this kind should at this stage of the 
pupil's progress be solved with diagrams. Solution of prob- 

1 \ Yz The answer is i\. 



lem 



12. How often can you find three fifths of one in one? 

1 % 



Solution : M 



13. How often can you find four fifths of one in one ? 
Prove it. 

14. How often can you find two thirds of one in one ? 

1 M 
Prove it. Solution: I^TTl oi:fpp=iKs. 



15. What is one half of five ? One third of five ? One 
fourth of five ? One fifth of five ? Solution of first : 



• i + i + i + i -Hi = 24. 

16. What are two thirds of five ? Three fourths of five ? 
Two fifths of five ? Solution of first : 

I I i^ 

17. I know a number to which if I add three it will 
make five ; what is it ? 

18. I know a number that contains two twos and one 
one ; what is it ? 

19. I think of a number to which if I add four it will 
make five ; what is it ? 

20. I know a number whose half is two ; what is it ? 



Suggestions for Teaching N^unibers. 105 

21. I know a number whose half added to it makes three; 
what is it ? 

22. From what number can I take its third and have two 
left? 

23. If to a certain number I add its two thirds, the sum 
will be five ; what is the number ? 

24. What number added to one and one half makes four ? 

25. I think of a number that is three less than five ; what 
is it? 

26. What number is two less than three ? 

27. If from a certain number I take its fourth, three will 
be left ; what is it ? 

28. What number doubled makes five ? 

29. Three times a number and twice the number make 
five ; what is the number ? 

30. What number doubled and its half added makes five ? 

31. What number increased by its fourth equals five ? 

32. What number lacks two of being five ? 

33. What two unlike numbers make five ? What unlike 
numbers make four ? What like numbers ? 

34. I buy apples at two cents each and they cost me four 
cents ; how many do I buy ? 

35. I bought candies at one cent each, and out of three 
cents received one cent change ; how many did I buy ? 

2,^. I paid four cents for two pencils ; how much did 
they cost me apiece ? 

37. I bought two-cent oranges, and out of five cents re- 
ceived one cent change ; how many did I buy? 

.38. Sarah had five two-cent pieces and gave one to each 
of three boys ; how many had she left ? How many cents 
could she get for them ? 

39. A bird laid five eggs and hatched them all but four ; 
how many did it hatch ? 

40. Two boys and three girls went coasting ; the number 
of boys is what part of that of the girls ? By what part of 



io6 Science and Art of Education, 

their number should that of the boys have been increased 
to have made it equal to that of the girls ? 

41. Jane has four ducks and three pigeons ; how many 
times as many ducks has she as pigeons ? The number of 
pigeons is what part of that of the ducks ? 

42. Henry's book cost him three cents, and his pencil 
one third as much ; how much did both cost ? The differ- 
ence in cost of the two is what part of the cost of the book ? 
The cost of the pencil is what part of the difference ? 

43. One is what part of two ? of three ? of five ? 

44. Two is what part of three ? of four ? of five ? 

45. Three is how many times two? Three is what part 
of three ? of four ? of five ? 

46. Four little birds were in a nest and one half of them 
flew away ; how many remained ? 

47. Three boys are playing ball ; what part of their num- 
ber must be added to them to make it five ? 

48. How many two-cent postage-stamps can you get for 
five cents ? 

49. How many yards of tape, at three cents a yard, can 
you get for five cents ? What part of a yard for two cents ? 

50. If your mother should send you to the store with five 
cents, to buy thimbles that cost four cents each, how many 
could you get ? How much change would you get ? 

51. With five cents buy all the two-cent lemons you can? 
How many can you get ? Prove it. 

52. A bushel of apples costs three quarters of a dollar ; 
how many bushels can you get for two dollars ? for three 
dollars ? For two and one half dollars ? 

53. A bushel of pears costs two dollars ; how many 
bushels could you get for five dollars? for two and a half 
dollars ? What part of a bushel could you get for one and 
a half dollars? for three fourths of a dollar? for a half 
dollar ? How many pecks could you get for one and one 
fourth dollars ? 



Suggestions for Teaching Numbers. 107 

54. A peck of plums costs a half dollar ; how many half 
bushels could you get for a dollar and a half. How many 
bushels for three dollars ? 

55. When a half bushel of quinces costs one and one half 
dollars, how much does a peck cost ? how much a bushel ? 
What part of a bushel could you get for a half dollar ? for 
two dollars ? 

56. When a gallon of honey costs two dollars, what part 
of a gallon can be had for a half dollar ? for a dollar ? 
How many quarts for a dollar and a half ? What part of a 
dollar would a pint cost ? 

57. At five dollars a yard, what part of a yard of cloth 
can be bought for three dollars ? what for two and a half 
dollars ? for one dollar ? 

58. A quince and a peach together cost three cents ; how 
much did each cost, if the quince cost twice as much as the 
peach ? 

59. Two oranges cost five cents ; how much did each 
cost, if one cost a cent more than the other ? 

60. Three times a number less twice the number equals 
one ; what is the number ? 

61. Once a number and half the number equal three; 
what is the number ? 

62. There are three times as many geese in a pond as 
ducks; if there are three geese, how many are there alto- 
gether ? 

67,^ Sarah has three canary-birds more than Anna ; how 
many have both, if Anna has one ? 

64. In three halves, how many ones ? Three halves 
equal what part of two ? 

65. In five thirds how i 1 1 1 1 

many ones? How many .,,... ,^ ,^ ^ P" ^ 

7873737373 

halves? , CKOG[7> ^Q^3^ ==^^ 

Remark. — The second ^ H H ' — ' I ' ' 

part of question No. 65 may be solved with the following 
diagrams : 



io8 



Science and Art of Education. 



66. In five fourths how many ones ? how many halves ? 
Solution of second by diagrams : 






\i 










1 




















1 


h 











H 



% 



67. One and one half is what part of two ? Solution : 

68. One and one fourth is what part of five ? Solution : 



69. How many thirds in one half ? Solution : 



70. One third is what part of one half ? Solution: 
One third contains two of the three equal parts of 
one half, and is therefore two thirds of it. 

71. How many fourths in one half? 

72. One fourth is what part of one half? 

73. How many times is one half in two thirds? 
Solution: Examining the diagram, we find that 
one half of it contains three blocks, and two thirds 
four; the problem, therefore, is reduced to finding 
the number of times three blocks are found in four blocks 

Ans. I J. 

74. One third is what part of two thirds ? 

75. Two fourths are what part of three fourths ? 

76. Three fifths are what part of four fifths ? 

77. One half is what part of three fourths ? 



Suggestions for Teaching Numbers. 



109 



J- 

4- 
5- 
6. 

7. 
8. 

9- 
10. 



Lesson on Six. 
a. What the Pupils Must Discover 



I. 5-1-1= II. 6—2= 21. f of 6= 



1+5 = 

4 + 2 = 

2-f4= 
3 + 3 = 
3x2 = 
2x3 = 
I x6= 
6-1 = 
6-3 = 



12. 

13- 
14. 



6-5 = 

6h-i = 

15. 6-f-2 = 

16. 6-^3 = 

17. 6-^4= 

18. 6-v-5 = 

19. 6-^6 = 

20. ^of 6 = 



23- 
24. 
25. 
26. 
27. 
28. 
29. 
30- 



iof 6= 
I of 6= 
iof6= 
I of 6= 



I of 6= 
iof 6= 
I of 6= 



32. f of 6= 

33. I of 6= 

34. \ of 1 = 

35. fof 1 = 

36. f of I = 

37. i of 2= 

38. i of 4= 
39- i of 5 = 
40. I of 2= 



41. I of 2= 

42. I of 4= 

43. I of 4= 

44. iof 5 = 

45. f of 5 = 

46. I of 6= 

47. I of 5= 



b. Examples of Seat-work. 



five -I- one = 
one + five = 
four -1- two = 
two 4- four = 
three -f- three = 
three x two = 
two X three = 



9- 
10. 
II. 
12. 

13- 
14. 



one X SIX = 

six— one. = 

six— three = 

six — two = 

six— five = 

six— six = 

six-f-one = 



15. six-^two = 

16. six -J- three = 

17. six -4- four =: 

18. sixH-five = 

19. six-7-six = 

20. one half of six = 



two halves of six = 

one third of six — 

two thirds of six = 

one sixth of six = 

two thirds of six = 

three thirds of six = 

four sixths of six = 

28. five sixths, of six = 

29. one fourth of six = 

30. three fourths of six = 

31. one fifth of six = 



21, 

22. 

23. 
24. 
25. 
26. 
27. 



32. 
33. 
34. 
35. 
36. 



three fifths of six 
four fifths of six 
one sixth of one 
three sixths of one 
four sixths of one 

37. one sixth of two 

38. one sixth of three 

39. one sixth of four 

40. one sixth of five 

41. three sixths of five 

42. five sixths of five 

etc., etc. 



' c. Suggestive Questions and Froblefns. 
I. Take five cubes. Take another. How many alto- 



Science and Art of Education. 



gather have you taken ? (You have taken six.) Five and 
one are how many ? 

2. Four and two are how many ? Three and two ? 

3. Three and three are how many ones? how many 
twos ? 

4. Six ones are how many twos ? how many threes ? 

5. How many fours in six ? 

6. Make six of ones ? how many does it take ? 

7. Make six of twos ? how many does it take ? 

8. Make six of threes ? how many does it take ? 

9. Make six in all the ways you can mentally, and in 
every case tell the result. 

10. Three twos are how many ones ? 

11. Two threes are how many ones ? 

12. Six ones are how many twos ? how many threes ? 

13. I know a number whose third is one; what is it ? 

14. 1 think of a number which doubled makes six; what 
is it? 

15. I think of a number whose half added to it makes 
six; what is it ? 

16. If to twice a number I add two, it makes six ; what is 
the number ? 

17. From what number can I take its half and have one 
left? 

18. What number diminished by its third leaves four? 

19. If to a certain number I add its fifth, it makes six; 
what is the number ? 

20. From what number can I take half of four and have 
four left ? 

21. If to a certain number I add a third of three, it 
makes two; what is the number ? 

22. Three times a number less once the number is four ; 
what is the number ? 

23. What equal numbers make six ? What unequal num- 
bers ? 



Suggestions for Teaching Numbers. 1 1 1 



24. How many fourths in a half? in a fourth and a 
half? . 

25. How many sixths in one? in a half? 1 

„ iU- J !> I — 1 — 1 






in a third ? 

26. A half and a third equal how many sixths ? 

27. A half, a third, and a sixth are how many ones ? 

28. Two thirds are how many sixths ? Three sixths 
equal how many thirds ? how many halves ? 

29. Four sixths equal how many thirds ? how 
many halves ? 

30. Five sixths equal how many thirds ? how many 
halves ? 



31. At two cents a yard, how many yards of tape could 
you buy for six cents ? 

32. How much money will pay for four apples at a half- 
cent each ? at a fourth of a cent each ? 

T,-^' If plums sell at 'a fourth of a cent each, how much 
would six cost ? 

34. Sarah can walk two miles an hour and Helen three ; 
]iOw long would it take each to walk to a town six miles 
distant ? If they should start together, how far apart would 
they be at the end of two hours ? If they should start to- 
gether but go in opposite directions, how far apart would 
they be in one hour ? 

35. Alice went to the grocery with six cents ; she bought 
pears with one third of them, cherries with one half of 
them, and chestnuts with the remainder ; how much money 
did she give for each ? 

36. If fish-hooks cost a half-cent each, how many can 
Charles buy for three cents ? 

37. Henry bought rabbits at a half-dollar each ; how 
many did he get for two dollars ? 

38. When hickory-nuts sell at four cents a quart, how 



1 1 2 Science and Art of Education. 

many pints can John get for six cents ? How many for two 
cents? 

39. Jennie and Cora went to the cellar for apples ; how 
many did each get, if Cora had twice as many as Jennie, 
and both had six? 

40. If a yard of muslin costs six cents, what does a foot 
of it cost ? 

41. One sixth is what part of three sixths? of two sixths ? 
of five sixths ? 

42. One sixth is what part of one ^ _^ 
half? Solution by diagram : h-M-f-H-H" 

43. One sixth is what part of one y^ 

fourth ? Solution by diagram : ^^\\ \ \ \ — \==y^ 

44. What part of one is one half of one ^ 

third ? Solution ; | | |- j one of the blocks, being a half of 
of a third, is a |^-^l I ' sixth of one, or of the whole. 

45. One half of one half is what part of one ? 
3-1 



H — V- 



or, 




r 










H 












L_ 



Solution. , , ^ — , — , V.., ., 

L_l l=M. 

46. One third of one half is what part of one ? 

\/ 
Solution : [— f 



= H. 



47. What is one half of two thirds ? of three thirds ? 



Remark i. — It is not deemed necessary to indicate what 
the pupils should discover, nor to give examples of seat-work, 
further than the number six, 

2. — From here on figures may gradually be introduced. 

Suggestive Questions and Problems on Seven. 

1. Put four cubes upon the table ; put three more on. 
How many ones have you put on ? (Four ones and three 
ones are seven ones.) 

2. Three twos and one are how many ? 

3. Five and two are how many ? 

4. Two threes and one are how many ? 



Suggestions for Teaching Numbers. 1 1 3 

Seven ones are how many twos ? how many threes ? 

6. What number is three less than seven ? 

7. What number added to five makes seven ? 

8. I think of a number whose double and one added 
make seven ; what is it ? 

9. What number is that whose half added to it makes 
six ? 

10. What is one seventh of seven ? What are three 
sevenths of seven ? five sevenths ? seven sevenths ? 

11. Two thirds of three apples and three fourths of four 
apples are how many apples ? 

12. Jane found one half of four chestnuts and Alvira two 
thirds of six ; how many did both find ? 

13. Of tape costing a half-cent a yard, Hannah bought 
one third of three yards, Eva three fourths of four yards, 
and Emily five sixths of six yards ; how much did they all 
pay ? 

14. QxiQ half of four and two thirds of six are how many 
ones ? how many twos ? how many threes ? 

15. Four sevenths of seven and three fifths of five are 
how many ones ? how many twos ? how many threes ? 

16. Make seven of twos ; how many does it take ? 

17. Make seven of threes ; how many does it take ? 

18. Make seven in all the ways you can mentally, and in 
each case state the result. 

19. How many two-cent postage-stamps could you buy 
with seven cents ? 

20. An orange costs three cents and a lemon two thirds 
as much ; how much do both cost ? 

21. How many three-cent and four-cent pies could you 
buy with seven cents ? 

22. How many pints in three quarts ? How many gills in 
a pint and a half ? 

23. How many feet in two yards ? in two-thirds of a 
yard ? 



114 Science and Art of Education. 

24. If two cakes cost six cents, what is the cost of each ? 

25. John has six pears which he wishes to share with his 
two companions ; how many will he give to each ? What 
part of them will he keep ? 

Remark. — A square has four equal sides and four right 
angles. When the sides are one inch, it is a square inch ; when 
one foot, a square foot ; when a yard, a square yard, etc. 

26. How many squares, an inch on each side, could you 
cut from a piece of paper two inches long and one inch 
wide? 

27. How many squares, two inches on each side, could 
you make from a piece of paper four inches long and one 
inch wide ? 

28. How many squares, a yard on each side, could you 
cut from a piece of muslin three yards long and two yards 
wide ? 

29. Henry borrowed a book from the town library, at a 
half-cent a week, and kept it until the hire of it amounted 
to three and a half cents ; how long did he keep it ? 

30. John has two three-cent pieces; how many quarts of 
milk, at two cents each, can he get for them ? 

31. Mrs. Smith borrowed Mrs. Brown's sewing-machine, 
at a quarter of a dollar a week, and kept it a month ; how 
much did she owe Mrs. Brown for the use of it ? 

32. Mr. Stone hires out his horse at the rate of three 
dollars for two days ; how much would he charge for the 
use of it one day ? How much for three days ? 

33. If two quarts of milk cost four cents, what will three 
quarts cost ? 

Suggestive Questions and Problems on Eight. 

I. How many ones in seven? How many twos ? How 
many threes ? How many fours ? How many fives .? Ho^y 
Jiiany sixes? How xnanj sevens.? 



Suggestions for Teaching Numbers. 1 1 5 

2. Put one to seven and how many have you ? (Seven 
and one are eight.) 

3. Place eight cubes upon the table and divide (separate) 
them into two equal groups. How many have you in each 
group ? What part of the whole ? 

4. Divide eight cubes into four equal groups and tell me 
how many you have in each group, also what part of the 
whole. 

5. Divide eight cubes into eight equal groups. How 
many have you in each ? what part of the whole ? 

6. See how many groups of three you can find in eight ; 
how many of five ; of sixe ; of seven. 

7. Five balls and three balls are how many balls ? 

8. Eight less two are how many ? 

9. Two and six are how many ? How many more than 
four ? How many more than seven ? 

10. Two fours are how many more than three twos ? 

11. One is how much less than one half of two ? 

12. What is the difference between three and two thirds 
of six ? 

13. See how many quarts of sand will fill this peck 
measure. 

14. What is the larger measure, a peck or a gallon ? 
what is the difference ? 

15. You may now find how many pecks of sand (or saw- 
dust) will fill this half-bushel. How many would fill a 
bushel ? How many half-bushels in a bushel ? 

16. How many pints of chestnuts, at four cents a quart, 
can you buy for six cents ? How many for two cents ? 

17. When milk sells at eight cents a gallon, how much 
would two quarts cost ? How much a pint ? 

18. Helen bought a yard and a third of six-cent muslin ; 
how much did she pay for it ? 

19. Margaret bought plums at four cents a quart, how 



1 1 6 Science a7id Art of Education. 

much did she pay for one fourth of a peck? for one 
eighth of a peck ? for a pint ? 

20. How many eighths in one half ? in one fourth ? in 
two halves ? in three fourths ? 

21. How many more eighths in two fourths than in one 
half? 

22. How many eighths in one half, one fourth, ^ 
eighth ? ^^- 1 I I 

23. Which is the larger, one half or one third ? L I I .- 
What is the difference ? 

24. Nora bought two thirds of a yard of lace at one 
store and one half of a yard at another ; how many yards 
altogether did she buy ? 

25. How many months in eight weeks ? 

26. Henry was hired to Mr. Carpenter at four dollars a 
month ; how much did he receive for three weeks ? for 
five weeks ? for three days ? for a week and a half ? 

27. Mr. Clay hired Mr. Thome's horse at the rate of 
eight dollars for six days ; he used the horse four days. 
How much did he owe Mr. Thorne ? 

28. Joseph borrowed five dollars of Philip, agreeing to 
pay a half-cent a month for every dollar borrowed ; if he 
kept the money two months, how much did he owe for the 
use of it ? 

29. Alfred rented two books from the library; the first 
was worth one dollar, the second two dollars. If he 
paid a cent a week for the first and two cents for the 
second, how much did he owe at the end of two and a half 
weeks ? 

30. One is what part of five ? of six ? of seven ? of 
eight ? 

31. Two is what part of six ? of seven ? of eight ? 

32. Three is what part of six ? of seven ? of eight ? 
2,2,. Four is what part of five ? of six ? of seven ? 
34. Five is what part of six ? of seven ? of eight ? 



Suggestions for Teaclmtg Numbers. 117 

35. Six is what part of seven ? of eight ? 

36. Seven is what part of eight ? Seven is how many 
halves of eight ? 

37. If you had one third of two yards of ribbon, what 
part of a yard could you make of them ? 

38. Three fourths of two quarts of peanuts would make 
how many whole quarts ? how many pints ? 

39. The surface of anything — a piece of paper or board, 
for example — is the outside of it. A square surface an inch 
on each side and all the angles of which are right angles, is 

called a square inch ; one foot on each side, a square 
foot, etc. A board three feet long and two feet 
wide contains how many square feet ? 

40. How many square inches could you cut from a piece 
of paper three inches long and three inches wide ? How 
many from one four inches long and two inches wide ? 

41. How many pieces of paper two inches long and one 
inch wide could be made from a piece three inches long 
and two inches wide ? 






^ 


M 


.1 


^ 



42. A piece of sheeting one yard long and two thirds of 
a yard wide contains how many square feet ? what part 
of a yard ? 

43. How many square feet in a piece of cardboard a 

half-yard square 

Remark, — A block all of whose surfaces are equal squares is 
a cube. If the sides of the squares are one inch the block is a 
cubic inch ; if they are a foot it is a cubic foot, etc. 

44. I have a cubical block of wood each of whose edges 
is two inches in length ; how many cubical J>locks whose 
edges are one inch could be made of it ? How many such 
could be made of a block two inches long and wide and 
one inch thick ? How many of one six inches long, one 
inch wide, and half an inch thick ? 



its 



Science and At't of Mducatioti. 



45. A bushel of cloverseed costs four dollars ; what is 
the cost of a peck ? of a quart ? 

46. One fourth is what part of three fourths ? 

47. Two fifths are what part of three fifths ? of four 
fifths ? 

48. One sixth is what part of five sixths ? of two sixths ? 

49. Three sixths are what part of four sixths ? of six 
sixths ? 

50. Four sevenths are what part of six sevenths? One 
half is what part of four sevenths ? 

51. One half is what part of five eighths? Three eighths 
equal what part of one half ? 

52. One fourth is what part of six eighths? of two 



thirds ? % 



53. One sixth is what part of one fourth ? Solution by 



diagram 



Md 


— 




— 




— 


— 













































=4r=% 



54. One seventh is what part of one sixth ? Solution by 



diagram : 



^( 






\-V7. 



55. One eighth is what part of one fifth ? Solution by 

hi 



. ;. 
















... 


rlionrfotn • 75 L 






v 
























































_ 



Suggestions for leaching Number 3. 119 

Suggestive Questions and Problems on Nine. 

1. Two threes and one two are how many? how many 
ones ? 

2. Three twos less two threes are how many ? 

3. Two fours less two thirds of six are how many ? how 
many twos? 

4. Which is the larger, four twos or two fours ? 

5. Four twos are equal to how many ones ? 

6. If to eight ones you add one one, how many ones 
have you ? (Eight and one are nine.) 

7. How many threes can you find in nine ones, or nine ? 

8. Nine contains how many twos ? fours ? fives ? 
sixes ? sevens ? eights ? 

9. How can you get one third of nine ? What are two 
thirds of nine ? 

10. Name the sums of the following numbers as quickly 
as you can : Three and four ; two and six ; five and three ; 
two and seven ; four and four ; one and four ; six and 
three; etc. 

11. Name the differences between the following num- 
bers : Three and five ; two and six; one and four ; two and 
seven ; five and eight ; four and nine ; one and eight ; 
seven and nine ; etc. 

12. From eight take two threes; from nine three take 
twos ? 

13. I know a number which doubled and one added 
makes this nine ; what is it ? 

14. Add one third of nine and three fourths of four. 

15. Into what like numbers can you divide eight ? nine ? 
six ? 

16. Three fourths of eight and one third of nine equal 
how many threes ? 

17. Make nine in all the ways you can mentally, and in 
each case tell the result. 



1 20 Science and Art of Education. 

1 8. In three yards how many feet? 

19. How many yards in five feet ? in eight feet ? 

20. What number is that whose third is three ? 

21. If to one third of a number I add two, the sum 
will be three ; what is the number? 

22. Three miles added to three fourths of the distance 
Samuel rode on his bicycle make nine miles ; how far did 
he ride ? 

23. If from four times the number of quarts of chest- 
nuts Henry gathered three be substracted, five will be left ; 
how many did he gather ? 

24. Five times the money Sarah received for making a 
dress added to two dollars equal seven dollars ; how much 
did she receive for her work ? 

25. Three fourths of eight cents is two thirds of the 
price of a gallon of milk ; what is the price ? 

26. It requires two thirds of nine yards of goods to make 
Ella a dress, and this is three fourths of the number re- 
quired to make one for Stella ; how many yards does Stella 
require ? 

27. Elmer is twice as old as Carrie, and the sum of their 
ages is nine years ; required the age of each. 

28. From four twos, take two thirds of nine, and what is 
the result ? 

29. How many square yards of carpet would be required 

to cover a floor nine feet long and six feet wide? What 

would be the cost of the carpet at a dollar and a half a yard ? 

Remark. — The pupils should by this time have learned that 
the contents of surfaces are obtained by multiplying the length 
by the breadth, and that the quotient of the contents divided 
by one of the factors gives the other. 

30. I want a piece of paper to cover the bottom of a box 
that is two inches wide and four and one half inches long ; 
how many square inches must it contain ? 

31. The leaves of my book contain eight square inches ; 
how long are they if they are two inches wide ? 



Suggestions for Teaching Numbers. 121 



32. I have a little cubical box whose inside is one inch in 
length, width, and depth ; how many half-inch cubes would 
fill it ? How many inch cubes ? Diagram, 

33. How many inch cubes would cover the bottom. of a 
box that is three inches long and two inches wide ? How 
many inch squares would cover it ? How can you ascertain 
without trying it ? 

34. How could you find the number of square inches on 
the cover of my book ? You may do it. 

35. Mr. Fox hired a carriage from the livery stable at the 
rate of three dollars for six days and used it four days; 
how much did he pay for the use of it ? 

2,6. Dora is working for Mrs. Wilson for a dollar and a 
half a week (of seven days); how much will she receive for 
four weeks and three and a half days ? 

37. How many two-cent pieces would you give me for 
■eight cents ? for seven cents ? six cents ? five cents ? 

38. As quickly as you can, give me the results of the 
following : Five less three ; eight less five ; seven less four ; 
eight less two ; six less three ; nine less six ; nine less four; 
seven less two ; three times two ; two times four ; three 
times three ; four times two ; eight less three ; five and 
four ; two and six ; twos in four ; threes in six ; fours in 
eight ; twos in three ; threes in four ; threes in nine ; etc. 

39. What part of one is one third of one third. Solu- 



H 



















tion I—,—,— 



40. What part of one is one half of one fourth : Solu- 



% 

















tion 

41. What part of one is one fourth of one half? Solve 
by diagram. 



122 



Science and Art of Education. 



42. One eighth is what part of one fourth ? Solve by 
diagram. 

43. Two eighths equal what part of one half. Solve by 
diagram. 

44. Three eighths equal what part of five eighths ? Solve 
by diagram. 

45. How many eighths in three fourths ? Solve by 
diagram. 

46. How many eighths in two thirds ? Solution by dia- 
gram. 

47. How many thirds in five eighths ? 

48. How many fifths in two thirds. Solution : 

% 



f- 



=3^. 



49. How many fourths in three fifths ? Solution : 



J 




r— ^ 




^ ^1 




^ - 





50. How many thirds in four ninths ? 

51. How many halves in two thirds ? Solution i 

% 



H 



-.i 



-litf. 



% 



H 



52. One third is what part of one half? 

53. How many eighths in one fifth ? Solution : 



'%. 



in 



Suggestions for Teaching Numbers. 



^n 



54. How many sevenths in one sixth ? Solution: 











■" 






















































































y. 

















n I 1 I M I M = v/a. 
55. One eighth is what part of one third? Solution : 



= %. 



56. One seventh is what part of one half? Solution: 



'%. 



Suggestive Questions a?td Problems on Ten. 

1. Three threes are how many ones? Add one to them 
and how many have you ? (Nine and one are ten.) 

2. Four twos and two ones are how many ones ? how 
many twos ? 

3. Make ten of fives ; how many does it take ? 

4. Two fours are how many less than ten ? how many 
more than six ? 

5. Six and how many make ten ? Four and how many 
make ten ? 

6. How many threes in ten ? How many fours ? How 
many sixes ? 

7. Seven and two are how many ? How many more than 
five ? How many less than ten ? How many more than 
eight ? 

8. How many must be added to two to make ten ? 

9. Make ten in all the ways you can mentally, and in 
each case state the result. 



1 24 Science and Art of Education. 

10. What is one half of ten ? One fifth of ten ? Three 
fifths of ten ? 

11. To one third of nine add three fourths of eight, and 
what is the sum ? 

12. What is the sum of three fourths of eight and two 
thirds of six ? 

13. What is the sum of two thirds of three and one half 
of nine ? 

14. Ten less seven are how many ? 

15. How many fifths of five must be added to five sixths 
of six to make nine ? 

16. I know a number which doubled makes ten ; what is 
it? 

17. I think of a number which taken three times and 
two added makes eight ; what is it ? 

18. What number diminished by its third leaves six ? 

19. What number increased by its half equals nine? 

20. Six times a number diminished by five equals one ; 
what is it ? 

21. The sum of two numbers is seven and one of them 
is four, what is the other ? 

22. What number added to one third of ten makes four ? 

23. If you double a number and take four from it you 
have six ; what is the number ? 

24. Ten thirds equal how many ones ? 

25. How many ones in ten fourths? in ten fifths? 

26. How many tenths in one fifth ? in one half? 

27. Two fifths and one half equal how many tenths ? 

28. Can you make tenths of fourths ? Why not ? 

29. What can you make of halves and fourths ? Why ? of 
halves and thirds ? Why ? of halves and fifths ? 

30. What can you make of halves, thirds, and sixes ? 

Solution by diagram : A^^^^^^i^^^ 
M }i % 



Suggestions for Teaching Numbers. 125 

31. One half and two thirds equal how many ones ? 

32. Can you add halves, fourths, and eighths ? How? 

33. Can you add halves and fifths ? How ? Show by 
diagram. 



34. In a piece of goods ten yards long and one yard 
wide how many square yards ? Prove it. 

35. How many yards long is a ten-foot pole ? 

2i(i. Helen's apron is two feet in length and a half-yard 
in width ; what part of a square yard of goods does it con- 
tain ? 

37. A board eight feet long contains four square feet; 
how wide is it ? What would its width be if it contained 
ten square feet ? 

38. My table is a yard in length and five sixths of a yard 
in width ; how much will it cost to cover it with oilcloth 
at six tenths of a dollar a square yard ? 

39. How would you find the number of square feet of 
paper required to cover the lower sash of that window ? 
Do it. 

40. Here is a chest that is four feet in length, two feet 
in width, and one foot in depth ; how many blocks, each 
containing a cubic foot, would exactly fill it ? How can 
you tell ? Show by a diagram. 

41. Henry bought a half-gallon of cherries at five cents 
a quart and paid for them with five-cent pieces ; how many 
did it take ? How many two-cent pieces would pay for 
them ? How many ten-cent pieces ? How many dimes ? 

42. John sold nine pigeons at the rate of a half-dollar a 
pair ; how much did he receive for them ? How many 
quarter-dollars ? 

43. When sugar is five cents a pound, what will one and 
cost two thirds pounds ? 

44. Mr. Down loans money at a third of a cent on a dol- 



1 26 Science and Art of Education. 

lar a month ; how much interest does he receive for the 
use of five dollars for three and a half months ? for six 
months ? 

45. At a half-cent a month on a dollar, in what time 
would six dollars give nine cents interest ? 

46. At a fourth of a cent a month, h'ow many dollars 
(how much principal) would in four months give eight 
and three fourths cents interest ? 

47. At what rate (how many cents on the dollar) will 
four dollars, in three months, give six cents interest ? 

48. If Henry adds the interest of nine dollars, at a third 
of a cent a month, for two and a half months, to the princi- 
pal, what will the amount (sum) be ? 

49. If five oranges cost nine cents, what is the cost of 
three of them ? 

50. When two fifths of a yard of tape cost four cents, 
what does a half-yard cost ? 

51. Three yards of lining cost nine cents ; at the same 
price, what would two and two thirds yards cost ? 

52. At two thirds of a cent each, how many oranges can 
be bought with two cents ? Solution: i. If they had cost 
one third of a cent each, for two cents six could have been 
bought ; but since they cost two thirds of a cent each, only 
half as many can be bought as if they cost one third 
of a cent each, or three. 2. Since two thirds of a cent 
pay for one orange, one third of a cent would pay for 
one half of it, three thirds for three halves, and two cents 
for two times three halves, which are six halves, or three. 

53. At three fourths of a cent each, how many yards of 
cord could you buy for two and a half cents ? 

54. Mary bought six dollars' worth of cloth at two and a 
half dollars a yard ; how many yards did she buy ? 

55. In ten pecks how many bushels ? In nine pecks how 
many ? 

56. In a half-bushel of cherries, how many gallons ? 



Suggestions for Teaching Numbers, 



127 



57. One is what part of ten ? of nine ? of seven ? of two 
and. a half? 

58. Five is what part of ten ? of nine ? of six ? 

59. What part of one is one third of one third ? 



60. What part of one is one fourth of one half? 

61. One half of one fifth is what part of one ? Solve by 
diagram. 











% 









62. How many fourths in two thirds ? ^4( 
(^2,' How many ninths in one half ? 



y 












■•- 


T 




. 1 





=4^. 



64. How many eighths in one third ? 



=2%. 



65. One fourth is what part of one third ? 



%. 



66. One fifth is what part of one half ? of one third ? 
Solve by diagram. 

67. How many fifths in one third ? Solve by diagram, 

68. One ninth is what part of one third ? 



128 



Science and Art of Education. 



69. One eighth is what part of one third ? 





... 




] 


1 1 I 










zri ' 




i li i i i i 



70. One fourth is what part of two sevenths ? 



M{ 



71. One half is what part of three fifths ? 

72. How many times can you find three fourths in six? 



73. How many times can you find two thirds in five ? 

74. How many times can you find one half in three 
fourths ? 

75. Two thirds are what part of four ? 



76. If a tailor can make a pair of pantaloons in a day 
how long would it take two tailors to make them ? 

77. Two boys can pick two quarts of berries in an hour , 
how long would it take one of them to pick them ? How 
long would it take three boys to do it ? 

78. Four boys gathered two bushels of chestnuts in two 
hours ; how many boys could, in the same time, have 
gathered three bushels ? 

79. If five cows can eat an acre of grass in one week, 
how long would it take one cow to do it ? How long 
would it take one cow to eat two acres ? 

80. Emma can make a dress in two days and Jennie in 
three ; what part of it can each make in a day ? What 
part of it could the two together make in a day? How 
long would it take them to make it working together ? 



Suggestions for Teaching Numbers, 129 

81. A man and a boy together do a piece of work in two 
days ; what part of it does each do, if the man does twice 
as much as tho boy ? How long would it take each alone 
to do it ? 

82. Bessie and Elsie received nine cents for picking 
strawberries ; how much did each receive, if Elsie received 
half as much as Bessie ? 

^2,. Henry can do three times as much work in a day as 
Samuel ; how long would it take Henry to do what Samuel 
can do in two days ? How long would it take Samuel to do 
what Henry can do in two days ? 

84. Find the length and width of the smallest board that 
you could exactly cover either with two-inch or three-inch 
squares. 

85. What is the smallest bag that you could make that 
could be exactly filled with a pint-measure or a quart- 
measure of chestnuts ? 

86. What is the length of the shortest pole that could be 
exactly measured either with a two-foot measure or with a 
yard-stick ? 

87. What is the smallest number of apples that you could 
all sell either by twos or threes ? 

88. What is the largest measure with which I could 
empty each of two boxes, one containing a quart of berries, 
the other a gallon ? 

89. Mr. Miller has two baskets of cherries, one contain- 
ing two quarts, the other three ; what is the largest cup 
with which he can exactly measure the contents of each 
basket ? 

90. What is the largest measure that is contained in three 
feet, six feet, and nine feet ? in four feet and eight feet .'' 
in four feet, six feet, eight feet, and ten feet ? 

I. Writing Numbers above Nine. 

Remark. — The pupils are supposed to have learned to write 
and use numbers up to ten. 



130 Science and Art of Education. 

1. They should now be told that nine is the highest 
number that we can write with one figure, and that above 
nine the numbers are written as tens and ones. 

2. Before the children are taught to write ten, they 
should have practice in finding the number of tens in a 
number of objects ; tying toothpicks, or other suitable 
objects, into bundles of tens, affords perhaps the best prac- 
tice in counting by tens. One bundle should be called one 
ten, two bundles two tens, three bundles three tens, etc. 

3. If a pupil has more toothpicks given him than make 
an exact number of tens, the remainder should be con- 
sidered as so many ones (or units). For example, if he 
should have sixteen given him, after having made all the 
possible bundles, he would have six toothpicks, or ones, left. 

4. To make the bundles, the children should sit or stand 
around a table, each having a small handful of splints or 
toothpicks before it, and rubber bands or threads to tie 
those of a bundle together. 

5. After the bundles have been made the children should 
count both them and the single things or ones, and write 
the results in columns prepared for the purpose. After each 
child has written its results or sums in the proper columns, 
the columns should be added, also the bundles and single 
things, and the results compared. 

Example i : Suppose there are four children and each has 
made its bundles ; the first having 2 bundles and 4 single 
things ; the second, 4 bundles and 3 single things ; the 
third, I bundle and 7 single things ; and the fourth, i 
bundle and 2 single things. They now write 
their results in columns, as here indicated, and 
add them. The first, or ones' column, gives i 
bundle and 6 single things. Writing the 6 under 
the ones' column and adding the bundle to the 
tens, gives as the result of both, 9 bundles and 6 single 
things. 



3 4 

4 3 

.1 7 

9 6 



Suggestions for Teaching Numbers. 131 

After the sums of the columns have been ascertained, the 
children should give their bundles and remaining tooth- 
picks to one of their number, who, after having made as 
many bundles as possible of the remaining single tooth- 
picks, should compare her bundles and remaining toothpicks 
with the sums of the columns. 

■Example 2 : Remark. — This example, besides carrying the 
work another step forward, also introduces the nought. 

Let us suppose that there are six children in the class, 
each having made its bundles ; the first having 3 bundles 
and 5 toothpicks ; the second, 4 bundles ; the third, 2 
bundles and 7 toothpicks ; the fourth, 3 j^^.j^^^ j^„, ones 
bundles ; the fifth, 2 bundles and 8 tooth- 
picks, and the sixth, 2 bundles. Adding the 
columns and also the toothpicks, it is found 
that there are ten bundles and eight bundles; 
but since there can be no more than ten of a 
kind, the ten-bundles must be tied together, 
making a ten-ten bundle, and its number written at the foot 
of a line next to the left of the tens. 

6. The children should work with toothpicks in connec- 
tion with figures until they can write and read numbers to 
one hundred at least. They should also be led to see that 
all numbers above ten are composed of tens or tens and ones; 
and instead of continuing to use the names tens and ones, the 
usual names should gradually be introduced. Thus, for ex- 
ample, instead of saying one ten and six, two tens and four, 
etc., they should learn the names sixteen, twenty-four, etc. 

Remark. — The nought, having no numerical value, is used 
to give the significant figures their proper places. 

7. The pupils should be led to see that the value of a 
figure depends upon the place it occupies in a number ; 
that, in- general, every ten of one place makes one of the 
next to the left, or higher, and also the reverse, namely, that 
Qvery place to the right is one tenth of the next to the left. 











3 


5 




i 







3 


7 




3 







2 


8 




3 





1 8 






132 Science and Art of Education. 

2. Suggestions on Teaching Numbers up to Twenty. 

The numbers up to twenty at least should be so thorough- 
ly taught that the pupils can instantly give the sum or pro- 
duct (less than twenty) of any two of them ; also the 
reverse ; and if a sum or product is given and one of the 
two numbers or factors that compose it, the other should 
instantly be upon the pupils' lips. 

3. A Device for Oral Addition. 

Two numbers below ten, whose sum exceeds ten, may 
readily be added by taking the difference between the larger 
and ten from the smaller, adding it to the larger, and add- 
ing the remainder of the smaller to ten, the sum. Example : 
To find the sum of 6 and 8. The difference between 8 and 
10 taken from 6, and added to 8, makes 10; and 4, the re- 
mainder of the smaller, added to 10, makes 14. 

Remark. — The teacher should lead the children to discover 
every device that will enable them to overcome their early 
difficulties and lighten their labor. 

4. The Four FimdaiJiental Processes Carried on Together. 
It should be borne in mind that addition, subtraction, 

multiplication, and division are to be carried on together. 

5. Pupils May Construct the Tables. 

To familiarize themselves with the relations of number, 
the children may. construct the tables, but in giving the 
results in class they must not think of the position of the 
numbers in the table, but must give them instantly — as 
far as reasonable, automatically. 

6. Suggestive Exercises for Seat Work. 



Remark.— 

1. 
I. 100 i = ii 


= what ? 
2. 
I. + 2=12 


I. 


3. 

1+0=13 


I. 


4. 

10 + 4= 


2. 9+0=11 


2. 12—0=8 


2. 


130 1 = 12 


2. 


1301 = 14 


3. 0-1 = 10 


3. + 3 = 12 


3- 


10 + 3=0 


3- 


14-0 = 9 


4. 84-3=0 


4. 1202=10 


4- 


13—0=9 


4. 


0^2 = 7 


5. 11-0=9 


5. 0x3=12 


5. 


+ 7 = 13 


5- 


11+0=14 



Suggestions for Teaching NtDubers. 133 



6. 


0-4-1 = 11 


6. 


12—7=0 


6. 13-5=0 


6. 7x2=0 


7. 


11—4=0 


7. 


705 = 12 


7. + 0=13 


7. 608 = 14 


8. 


+ 5 = 11 


8. 


0x0=12 


8. 0—0=3 


8. 0x0=14 


9- 


7+0=11 


9. 


0—0=7 


9. 13—8=0 


9, + 0=14 


10. 


+ 0=11 


10. 


+ 0=12 


10. II 2 = 13 


10. 0-0=5 


11. 


0—0=6 


II. 


0^0=6 


II. 13—0=6 


II.. o-^-o=2 




etc. 




etc. 


etc. 


etc. 




5. 




6. 


7. 


8. 


I. 


0+14=15 


I. 


0+10=16 


I. 10 + 7 =0 


I. 0-1 = 17 


2. 


15—0 =10 


2. 


16—0 =9 


2. 0—6 =11 


2. 1+0=18 


3. 


906 =15 


3- 


+ 3 =16 


3. 1708 =9 


3. 1803 = 15 


4. 


5x 0=15 


4. 


16—10=0 


4. 15+0 =17 


4. 15 + 3=0 


5- 


15—10=0 


5- 


0+11 = 16 


5. 17-3 =0 


5. 0-5 = 13 


6. 


1503 =5 


6. 


160 13=3 


6. 0—14=3 


6. 6 + 0=18 


7. 


o-i =14 


7. 


+ =16 


7. 17012=5 


7. 18-7=0 


8. 


0x0 =15 


8. 


0x0 =16 


8. 7+0 =17 


8. + 0=18 


9- 


+ =15 


9- 


0—0 =5 


9. + 9 =17 


9. 0x0=18 


10. 


oH-o =3 


10. 


O-T-O =4 


10. + =17 


10. 0—0=7 


II. 


0—0 =4 


II. 


8 + 8 =0 


II. 0—0 =6 


II. 0-7-0=2 




etc. 




etc. 


etc. 


etc. 




9. 




10. 


n. 


12. 


I. 


+ 3 =19 


I. 


19+0 =20 


I. i of 4=0 


I. iofo =f 


2. 


19-0 =15 


2. 


20 — 3 =0 


2. f of 8=0 


2. 2x0 =1 


3- 


11+8 =0 


3- 


0+14=20 


3. \ of 6=0 


3. of 6 =4 


4. 


1906 =13 


4- 


200 11=9 


4. 1 of 9=0 


4.0 X f =2 


5- 


+ 7 =19 


5. 


3+0 =20 


5. i of 5=0 


5. i of =i 


6. 


19—0 =6 


6. 


4+16=0 


6. \ of 4=0 


6. f of 2 =0 


7. 


3+16=0 


7- 


20—8 =0 


7. i of 8=0 


7. i of 3 =0 


8. 


1908 =11 


8. 


+ 6 =20 


8. 1- of 9=0 


8. 1 of 4 =0 


9- 


0-14=5 


9- 


+ =20 


9. f of 7=0 


9. 1 of 10=0 


10. 


0+0 =19 


10. 


0x0 =20 


10. t of 4=® 


10. f of 2 =0 


II. 


0-0 =8 
etc. 


II. 


o-=-o =5 
etc. 


II. \ of 7=0 


II. 1 of 6 =0 




13. 




14. 


15. 


16. 


I. 


4 = 1 of 


I. 


4 X i=o 


I. 9= f of 


I. f of 12=0 


2. 


5 = i of 


2. 


X 4=5 


2. 8= f ofo 


2. f of 3 =0 


3- 


6 = f of 


3. 


f of 0=3 


3- 3= lofo 


3. f of 4=0 


4- 


X 1 = 1 


4. 


i of o=i 


4. 2= 5 X 


4. 1 of =1 



1 34 Science and Ar't of Education, 



5- 


1 of o = 3 5. 


X ^ = 2 5. 12= f 


of 5. X f = j 


6. 


8 = f of o 6. 


\ X 0=^ 6. 1= f of 6. i of =7 


7. 


9 = f of o 7. 


|ofo=i 7. 5= 3 


X 7. i of =3i 


8. 


2 = f of o 8. 


X 2=3 8. i= i 


X 8. J of 2 =0 


9- 


1=3x0 9. 


2 X 0=5 9. I— 


= i 9. 2 X =i 


lO. 


6 = f of 10. 


1 of 5=0 10. + 3J 


= 5^ 10. X ii=4 


II. 


t — -i of II. 


xf=| II. I05 


= 2 II. iof =f 




17. 


18. 


19. 


I. 


i of = i 


I. iofi =0 


I. 5 = f of 


2. 


i^ X = 4 


2. f of f =0 


2. f of 2 =4x0 


3- 


X 2 = if 


3. i of 1=0 


3. f of 3 =3x0 


4- 


|ofo=i4 


4. 1 of 4=1 of 


4. 18 = I^ X 


5- 


o-i-4 = | 


5. |of 5=2 X 


5. 2 X = -^ of 3 


6. 


li xo=5 


6. li X 2i=^ofo 


6. ^off = iofo 


7. 


f xo = 3 


7. i= fofo 


7. 10 =2 X ^Ofo 


8. 


I J X 0= I 


8. \Qi f=o 


8. 1 =1 of 


9- 


2| X = ^ 


9. -1 = f of 


9. \q{\ = ^ofo 


lO. 


3i = i of 


10. 1 = 1 of 


10. ^ of ^ = ylj of 


II. 


2i X = i 


II. f = i of 


II. 18 =41 X 




20. 


21. 


22. 


I. 


9 = 9xi.of 


I. 12 -f-O =9 


I. OH-4i =5i 


2. 


1 of 2=f of 


2. 9 = 12 X 


2. 5 = 1 ofo 


3. 


2j X 0=18 


3. 10 = 1 of 


3. 1 of =9 


4. 


X 1=15 


4. 15 = 16 X 


4. X 18 =17 


5- 


f of 2=1 X 


5. i of 5 = loxo 5. 14 = 15 X 


6. 


f of 2=j of 


6. 0x9 =6 


6. 1 5 = 1 of 


7- 


iof |=iofo 


7. 5 -=- = 20 


7. 13 = i of 


8. 


% of i=f of 


8. 8 -f-o = If 


8. 9x1^=0x2 


9- 


f of 4=16 XO 


9. f of =2 


9. 3l ^ 1 =0 


lO. 


17 = 5f XO 


10. 6x0 = i\ 


10. 4^ -^ =3 


II. 


f 2 = lf 


II 1 X = i 


II. f -f-o = ^ 




23. 


24. 


25. 


I. 


0-^4 =li 


I. 0x18=19 


I. 2i -^o=i 


2. 


3-^0 =^ 


2. 4-5- 0= 8 


2. f -^o=ii 


3- 


6of =4 


3. 5- f = .0 


3. -|=i 


4. 


102 =^ 


4. o-T- \-= 6 


4. il +o=3i 


5- 


foa =1^ 


5. f + 0= 1 


5. 2i Xi=0 


6. 


o-i-f =1 


6. 13 X 0=17 


6. i/^ xo=i7 



Suggestions for Teaching Numbers. 



135 



7. f-O : 


=3 


7- J 


2-7- O: 


= 9 


7. f 


oi = 


=3 


8. O-f : 


=2 


8. 


OX 1 


=8i 


8. f 


=5x0 


9. f Xi : 


=0 


9- 


9= OX 12 


9. 2f 


X2 = 


=1 of 


10. 2i— : 


= ii 


10. 


I5H- 


=20 


lo- ^ 


— 0= 


=f 


II. 0-^4ir 


=4i 


II. 


0- i: 


= 15 


II. li 


+ 0= 


=3i 


26. 






27. 






28. 




I. o-i| 


=2f 


Lemons 


. cts. Lemons. 


Oranges 


cts. 


Oranges. 


2. 12 -f- 


=31 




2 


3 




4 


5 


3. oofii 


=2 




3 


2 




2 


7 


4. 3 -f-O 


=8 




I 


4 




3 


5 


5. 4-f-o 


=3 




4 


3 




6 


3 


6. 9x0 


= 19 




2 


4 




7 


2 


7. X 17 


= 13 




5 


3 




6 


2 


8. 7^0 


= 12 




2 


6 




8 


3 


9. II X 


= 16 




6 


2 




15 


2 


10. 12 — 


=8| 




4 


4 


2 


4 


3 


II. fofl 


=0 




5 


2 


3 

2 


9 
6 


4 
3 


29. 






30. 






31. 




Apples, cts. 


Apples. 


Yds 


cts. 


Yds. 


Yds. 


cts 


Yds. 


3 12 


4 


4 


12 


6 


1 


15 


i 


4 16 


3 


6 


12 


9 


f 


3 


2i 


2 10 


4 


8 


16 


i 


li 


2 


3 


5 15 


4 


5 


15 


i 


2i 


4 


1 


2 4 


8 


2 


16 


i 


f 


li 


1 


7 7 


5 


i 


2 


3 


f 


li 


2i 


4 20 


3 


i 


2 


i 


4 


If 


4 


6 12 


7 


I 


2^ 


4 


2 


2i 


f 


7 14 


9 


f 


8 


i 


1 


2 


f 


9 18 


8 


f 


9 


1 


i 


f 


2i 


8 16 


7 


2 


16 


f 


li 


2i 


2f 



Remark. — To indicate how the exercises under 27-31 should 
be read, the first under 29 may be taken as an example. If 3 
apples cost 12 cents, what cost 4 apples } 

7* Suggestive Problems for Oral and Seat Work. 

1. If 4 quarts of milk cost 8 cents, what will 8^ quarts 
cost? 

2. If 2\ yards of tape cost 15 cents, what will i|- yards 
cost. 



136 Science and Art of Education. 

3. My table is 3-i feet in length and two feet in width ; 
how many square feet of oilcloth wdll cover it ? 

4. A piece of cloth is one yard in length and the same 
in width ; how many square feet of paper would cover it ? 
How many square feet does it contain ? 

5. How many square feet in a surface a foot square ? in 
one a yard square ? in one 3 feet square ? in one 4 feet 
square ? 

6. How many square yards in a surface 6 feet long and 
3 feet wide ? 

7. I have a box that is 2 feet in length, i foot in width, 
and li feet in heigl^it ; how many square yards of paper 
would cover it ? 

8. Show me an inch on this foot-measure. How many 
inches in length is the whole measure ? How many inches 
are in a foot ? in a ^ foot ? in f of a foot ? in f of a foot ? in 
I of a foot ? in i^ feet ? in i^ feet ? 

9. How many feet in 9 inches ? in 8 inches ? in 14 inches ? 
in 18 inches ? in 3 inches ? 

10. How many quarts in 2 pecks ? in a half-bushel ? 

11. How many pecks in 12 quarts? in 18 quarts? in 6 
quarts ? 

12. At 8 cents a gallon what would 12 quarts of milk cost ? 
What would 18 quarts cost? 

13. Take the weights and find out how many ounces 
make a pound (avoirdupois). How many ounce weights are 
as heavy as f of a pound ? What part of a pound weighs 
as much as 12 ounces ? 

14. How could you add into one sum 2\ feet and 4 inches? 
How 5 feet, f of a foot, and 8 inches ? How 3 yards, 2f 
feet and 18 inches ? 

15. Take the yard-stick and measure 5^ yards from the 
platform through the middle of this aisle. How many feet 
did you measure ? 5^ yards, or 16^ feet, are called a rod. 

16. How many rods in the length of this room ' How 



Suggestions f 07- Teaching Numbers. 137 



many in the width ? How many feet in a ^ rod ? in ^ of 
a rod ? in f of a rod ? 

Remark. — The pupils should make inch, foot, yard, and rod 
measures and use them in measuring lengths and distances. 
The shorter measures maybe made of wood, the longer of twine 
or cord. 

17. Add into one sum i rod, i^ yards, i foot, and 8 
inches ? 

18. Find the sum of 3 gallons, 2 quarts, and i pint. 

19. Find the sum of f of a pound and 7I ounces. 

20. How many pints in f of a peck, 3I quarts, and f of 
a pint ? 

21. How many feet in ^ of a rod, 2I yards, 2 feet, and 9 
inches ? 

22. How many ounces in | of a pound and 2% ounces ? 

23. In 15 pints how many gallons? One pint is what 
part of a gallon ? of 3 quarts ? Three quarts are what 
part of three gallons ? 

24. In 18 inches how many feet ? What part of a yard ? 

25. Nineteen ounces equal how many pounds ? 

26. Twelve ounces equal what part of a pound ? 

27. In 17 feet how many yards ? how many rods ? 

28. How many feet in 16 inches ? in 10 inches ? 

29. In 15 weeks how many months ? in 17 weeks ? 

30. How many weeks in 2\ months ? in ^ of a month ? 

31. How many days in 2^ weeks ? in f of a week ? 

32. In 14 quarts how many pecks ? what part of a 
bushel ? 

33. In 20 pints how many pecks ? what part of a bushel ? 
how many half-bushels ? 

34. How many months in i\ years ? in if years ? in f 
of a year ? 

35. How many years in 15 months ? in 8 months ? in 9 
months ? 

36. Find the interest of $2^ at 6 per cent (6c. on a dollar 



138 Science and Art of Education. 

for a month) for f of a year? for 16 months? for 3 
months ? 

37. What is the interest of $4! at 3 per cent for i^ years ? 
for 4 months ? 

38. The interest of $5 for 2 years is 20c.; what is it of 
$1 for I year? 

39. What principal, at 2c. on a dollar a year, will in 16 
months give 15c. interest ? 

40. In what time (how many years) will $6, at 2\ per 
cent, give i8c. interest ? 

41. If by selling my knife for i6c. I gain \ of the cost, 
what was the cost ? 

42. Sarah lost \ of its cost by selling her bird for 12c.; 
how much had she paid for it? 

43. By selling a book for loc. I lost \ of its cost ; what 
should I have sold it for to have gained \ of its cost ? 

44. One of two boys can do a piece of work in 3 days, 
the other in 4 days ; what part of it can each do in a day ? 
What part can both together do in a day ? How long would 
it take both together to do the whole of it ? 

45. If it takes A 2 days to do a piece of work and B 5 
days, how many days would it take them together to do it ? 

46. Four men can do a piece of work in 2 days ; how long 
would it take one of them alone to do it ? Solution : 
00 + 00 + 00 + 00 — 8. Let every o represent a day's 
work of each man ; then all of them will represent 8 days' 
work of a man. 

47. If 5 girls can make a certain number of dresses in 4 
days, in how many days could two of them do the same 
work ? Illustrate by diagram or other form. 

48. If 5 boys can pick 30 bushels of apples in a day, how 
long would it take i boy to do it ? How long would it 
require 3 boys to do it? 

49. If 10 is f of a number, what are f of it ? 



Suggestions for Teaching Numbers. 139 

50. If \ of Henry's ducks equal -f of his turkeys, and he 
has 24 ducks, what is the number of his turkeys ? 

51. A watch and chain cost $15 ; what was the cost of 
each, provided the chain cost f as much as the watch ? 

52. If Jennie adds 8 years to \ of her age, the sum will 
be her age ; how old is she ? 

53. One half, ^, and i of a certain number added to 8 
make 21 ; what is the number ? 

54. At f of a cent each, how many lemons can be bought 
for 4 cents ? 

55. Alberta bought 4 oranges for 3 cents ; what was the 
cost of each ? How many did she get for i cent ? 

56. At $f a yard, how many yards of cloth can be bought 
for $6 ? 

57. A boy gave $3 to a number of beggars, giving to each 
$f ; how many beggars were there ? 

58. If to Alfred's money you add 4 times his money, he 
will have 25c.; how much money has he ? 

59. Find the difference between a square inch and an 
inch square ; between 2 square inches and 2 inches square. 

60. What is the difference between a foot square and a 
half-foot square ? Illustrate with diagram, 

61. How many 6-inch cubes could you make of a cubic 
foot ? how many 4-inch cubes ? How many 2-inch cubes 
could you make of a 6-inch cube ? Diagram. 

62. How many cubic feet in a box that is 3 feet long, 2 
feet wide, and i foot deep ? 

63. Could you find how many square yards of carpet 
would cover this floor ? How would you do it ? If the 
carpet were f of a yard in width, could you find how long 
a piece it would require to cover the floor ? Illustrate with 
diagram. 

64. How could you find the number of square feet of 
paper required to cover the walls and ceiling of this room ? 
Could you find the number of square yards ? How ? Could 



146 



Science and Art of Education. 



you find how long a roll it would require if the paper were 
only a half-yard in width ? 

65. How many foot cubes would cover the floor of this 
room ? how many would fill the room ? How many cubic 
feet in this room ? how many cubic yards? 



8. 



Diagrams and Figures. 



1. The teacher should be careful that pupils do not mis- 
take fractional expressions for fractions ; fractions are parts 
of things or wholes, and fractional expressions, as ^, ^, etc., 
are the signs or language by means of which they are repre- 
sented. 

2. Fractions may be introduced and to some extent 
taught by means of folding paper Oi by various forms of 
diagrams. The following illustrations, in addition to those 
already given, may prove helpful to teachers, and may be 
preferred by some. 



%/ 
/% 


k 










M 




M 




M 




M 




3. After pupils have learned to work with real fractions 
(parts of objects) they should be taught the signs by which 
they are represented. 

9. Adding^ Subtractings Multiplying, and Dividing by 

Diagrams. 
I. How many fourths in i ? in i and \ ? What >^ M 



part of one is i of i ? i + i =? i -f J =? i + 
\ =? }+i + =? i-i =? i - 1 =? etc. 



^ 


^ 


M 


y\ 



2. How many sixths in one ? in ^ ? in \ ? What part of 



Suggestions for Teaching Numbers. 



14 



Ke 


Ke 


Ke 


Ke 


Ke 


Ke 



I is J of J ? 4 of I ? J- is what part of i ? i is Vi. M_ 
what part of J ? -J is what part of -J ? How many ^ 
sixths in ^ ? How many ^ in | ? How many j^ 

*-i=?f-i=?t- i =? i - i =' * ^ -i =? etc. 

3. How many ^ in i ? in i ? in h ? ^ is what part of ^ ? 
i of f ? i of J ? How many -J in f ? How 
many i in f ? What part of \ \% \ oi \} \ of 
\ ? How many i in | ? How many times are 
f contained in i ? ^ + i =? i + i =? f + i y. 



HHHy4 



+ i + i=^ f + I-? i-i-? 



t 



^ 


^ 


% 


Vs 


^^ 


Va 


Vs 


% 



= H^i 



l-l 



;? T -4- 



etc. 



? _j 

• 10 



Kg K9IK9I 


^ 


Ko 


Ko 


V^ 


H 


H 



4. i of -^ is what part of i ? How many ^ in ^ ? ^ 
is what part of ^ ? how many ^ in ^ ? ^ con- %\iys 
tains how many ^ ? how many |^ in f ? i + i =? 

==? etc. 

5. i of i is what part of i ? How many ^ in J ? how many 
-^ in I ? How many yV i'^ ^^ ^ ^^^ many -^ in H % 
1^ ? ^ is what part of | ? -^tt is what part of | ? 
How many yV i^ i •'' Mow ma.iy -J^ in :j- ? How 
many times is -^ contained In -fj; . How many 
times are ^ contained in |- ? -{q- are what part 



Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 


Ho 



=H 



■5 



6. .i + .i=?.2 + .i 
— .z =?4 — .i=? ., 



?i + 



t--4 



— ? 



I = 



Vs % 



= ? .2 + i=?.4 + .2 = ? 

.8-^1=? etc. 
























7. What is i of 1 ? i of i ? ^- of i :> How many yV in 



142 



Science and Art of Education. 



J ? How many 



\y^\ 



1^ ? in^ ? in i ? in g . ax^jw mciii^y g ^" s ' 

in i ? How many i in ^ ? } is what part 

of 1 ? i is what part of J ? |^ is what part % 

of I? How many^ in f ? i is what part 

of i ? How many J in f ? How many i in ,^.^ ,^.^ ,^^ ,^^ ,^^ ,^^ 

I ? I equal what part of | ? i + J + iV I Jj I I 

i - A -? i X -^ =? i X i =? I X i =? I X f =? I -^ i 

■i 3 . 2 ^ 1 . 1 "> 1 _• i -> irifi- 



^ 



Ke 


1 — i^ 
He 


He 


He 


He 


He 


He 


He 


He 


He 


He 


He 


He 


He 


He 


He 



X{r=? I X tV =•'' etc. 
9." What is i of 1 ? i of i ? i of i ? J of i ? i + ^ + i = 




lo. One fourth is what part of 
^^ is what part of | ? of -J ? ^V 
is what part of -^^g- ? of i ? i X 3*^ M 

=?ix S=?ixi=?fx^=? 

iXyV^? |Xf =?AXf =? 

=? A + t\ + *=?*-!-? A 



— ig- — ' 3 • 8 — ; etc. 




Suggestions for Teaching Numbers. 



143 



II. How many .01 in I ? How many .1 in |? in i ? How 
many .01 in il ? in i ? How 
many .1 in .50 ? -J- -f -^ + -oi 

tV=?.iX i.=?.oiX.i = ?iV 
X .i = ?ixi=?i XtV=?.2o 

X .10 — ? T^ + .3 =? .1 + -I 



100 

==? .1— .01 == 

-.01 =:?i-r 



= ? .01 + .4 



: J__ =r ? 

• 100 

=:?.I2 + 



.07 
?.l8 + .24 =? 

? 





To^o" 



.06 



.1 


.1 


.1 


.1 


.1 


.1 


.1 


.1 


.1 


.1 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 


.01 



=- -3- 
? etc. 



?.ii-f-.o5 =? -56 -.4 



= ?.4 + 
.7 + .6 - 

.02=? .50-^.05 :=^i .75 
.09— .5 =?.12i + .14 : 

Remark.— If the foregoing work has been well done, the 
pupils will, to such an extent, have learned to make their own 
discoveries — to help themselves — that they will experience 
com^varatively little difficulty with the higher numbers and 
their combinations. 

10. Me. ial Addition of Two-figured Numbers. 

1. Two numbers, each composed of tens and units, may 
readily be added mentally by first finding the sum of the 
tens and then adding to it that of the units. Example : 
Find the sum of 28 and 39. Two tens and three tens are 
five tons, or fifty, and 8 units and 9 units, or 17 units, 
(added) equal sixty-seven. 

2. As soon as the pupils are far enough advanced to do 
so, instead of saying two tens and three tens are five tens, 
they should say twenty and thirty are fifty, etc. They 
should also be led to see that the same combination of 
units will invariably give the same units' figure. For 
example, 6 and 7, 16 and 17, 26 and 37, and so on, will all 
have 3 for the units' figure. 

1 1 . Subtracting a Greater from a Less. 

I. Subtracting one number from another when some of 
the figures of the minuend are smaller than those of the 



144 Science and Art of Education. 

subtrahend to be taken from them, may be developed by 

453 

means of tooth-picks. Example : —. Since 4 cannot 

219 

be taken from 3, one of the ten-bundles must be opened, 

its contents put to the 3 ones and the 4 taken from the 

sum. 

2. The same thing may also be developed with dollars, 
dimes, and cents of toy-money. Taking the foregoing 
example, since 4 cents cannot be taken from 3 cents, one 
of the ten-cent pieces of the minuend must be exchanged 
for cents and the latter added to the three to make the 
subtraction possible. One ten having thus been taken 
from the 5 of the minuend, 4 remain, from which the 3 of 
the subtrahend must be taken. 

3. The teacher should lead his pupils to see that finding 
the difference between two numbers is the same as finding 
what must be added to the smaller to make the larger, and 
that the difference may therefore be found either by addi- 
tion or subtraction. 

4. Methods of proof should also be developed, so that 
the pupils may have the means of testing the correctness 
of their work. 

Remark, — i. As far as possible and as long as necessary the 
solution of every problem should be illustrated either with 
objects, diagrams, or drawings. The drawings, at first crude, 
will with the teacher's assistance gradually improve and will 
create an interest in that kind of work. 

Remark. — 2, It cannot be too strongly impressed upon the 
mind of the teacher that a part of every lesson should be a 
review of some of the previously prepared lessons or subjects 
passed over. Daily reviews give not only clearness to concepts, 
but impress them tirmly upon the mind. 

12. Mental Multiplicaiion. 

I, Mentally finding the product of any two numbers up 
to 100, besides affording a good exercise for the memory, 
enables the pupils to perform many arithmetical operations 



Suggestions for Teaching Kunihcrs. 145 



without resorting to pencil and paper. To illustrate the 
method of doing this when both factors are less than 20, let 
it be required to obtain the product of 14 by 14. Fourteen 
is the sum of 10 and 4. First, multiply 14, the multipli- 
cand, by the 10 of the multiplier (or simply annex a 
nought) ; next the 10 of the multiplicand by the 4 of the mul- 
tiplier ; add the two products ; finally, multiply the 4 of the 
multiplicand by the 4 of the multiplier and add the prod- 
uct to the sum of the previous products. Operation : 
14 X 10 = 140 ; 10 X 4 = 40 ; 140 + 40 = 180 14X4 
= 16 ; 180 + 16 = 196. 

2. The foregoing method applies as well to unequal as to 
equal factors. For example, let it be required to find the 
product of 19 by 15. Operation : 19 X 10= 190: 10 X 
5 — 50 ; 190 + 50 = 240 ; 9 X 5 = 45 ; 240 -f 45 = 285. 

3. To find the product of any two factors that are 

greater than 20 and less too, multiply the multiplicand by 

10, the product by the number of tens in the tens of the 

multiplier, the tens of the multiplicand by the units of the 

multiplier; add product to preceding produt; multiply the 

units of the multiplicand by the units of the multiplier, and 

add product to preceding sum of products. For example, 

multiply 48 by 36. Operation .• 48 X 10 = 480 ; 480 X 3 

= 400 X 3 + 80 X 3 = 1440 ; 40 X 6 = 240 ; 1440 -j- 200 

+ 40 = 1680 ; 8 X 6 = 48 ; 1680 + 40 + 8 = 1728. 

Remark.- As an aid to memory, whenever it is possible, 
round numbers (as in the foregoing operation) should be mul- 
tiplied and added. 

13. Properties of Nine and Their Application. 

1. If a number is divisible by 9 the sum of its figures is 
divisible by nine. Illustrative examples : 18, 27, 63, 81, 
4896, etc. 

2. The remainder, or excess, of the division of a number 
by 9 is the same as that of the sum of its figures divided 
by 9. Illustrative example : 5843 divided by 9 leaves 2 as 



146 Science and Art of Education. 

the excess, and 20, the sum of its figures (5 + 8 + 4 + 3), 
divided by 9, leaves the same. 

3. The excess of the division of the sum of two or more 
numbers by 9 is the same as that of the excess of the sum 
of their excesses. 

From this fact is derived one of the simplest and quick- 
est methods of proving addition and subtraction. 

4. Application to Addition. — Rule : Find the excess of 
each of the numbers added, add these excesses, find the ex- 
cess of their sum, and if the latter is the same as the excess 
of the sum of the numbers added, the work may be con- 
sidered correct. Illustrative example : 

5873 5, excess of division by 9. 

4652 8. " " " 

8763 6. " " " 

"9 ^ 

19867.. .4. 22.... 4. 

The sum (22) of the excesses, divided by 9, leaves 4 as 
the excess; and the sum (19867) of the numbers divided 
by 9 leaves 4, the same excess. 

5. Application to Subtraction, — Rule : Find the excess of 
the minuend, subtrahend, and remainder or difference, 
add those of the remainder and of the subtrahend, and if 
their sum, or the excess of their sum, is the same as that of 
the minuend, the work may be considered correct. Illus- 
trative example : 

8975 2, excess. 

7487.... 8^ " 

1488 3, " 

Adding (3) the excess of the remainder and (8) that of 
the subtrahend and dividing the sum by nine, the excess 
found is 2, the same as that of the minuend. 

Note. — The excess of the division of the product of two or 
more factors by 9 is the same as the excess of the product of 
the excesses of the factors. 



Suggestions for Teachmg Numbers. 147 

From this fact is derived one of the simplest and most 
expeditious modes of proving multiplication and division. 

6. Application to Multiplication. — Rule : Find the excess 
of the product and of each of the factors, multiply that of 
the multiplicand (or the reverse) by that of the multiplier, 
find the excess of their product, and if it is the same as 
that of the product of the factors, the work may be re- 
garded as correct. Illustrative example : 

5873----5 
687.. ..3 



411U 15. . . .6 
46984 
35238 



4034751 6 

The excess of the product (4034751) is 6, that of the 
multiplicand (5873) 5, and that of the multiplier 3 ; and 
the excess of the product (15) of the latter two is 6, the 
same as that of the product of the factors. 

7. Application to Division. — Rule: Find the product of 
the excess of the divisor and of that of the quotient ; to it 
add the remainder, or its excess (if it has one) ; and if the 
excess of the sum is the same as that of the dividend, the 
quotient may be considered correct. Illustrative example : 
26) 6547 (251 
52 

134 
130 

47 
26 

21 
The excess of 9's of the divisor (26) is 8^ that of the 



1 48 Science and Art of Education. 

quotient (251), 8, their product is 64 ; to this add (3) the ex- 
cess of the remainder (21), and the excess of the sum (67) 
is 4, the same as that of the dividend. 

Note. — The only case in which the foregoing proofs could 
fail would be when one error would balance another — when, 
for instance, 56 would be written 65, or o written instead of 
9; but since such errors are of the rarest occurrence, the proof 
by the rejection of the 9's may be considered of equal validity 
with any others. 

14- Co?nposition of Nu?nbers. 

1. The pupils should be led to see how numbers are com- 
posed. The following may serve to show how it may be 
done : 

78964 = 70000 + 8000 -f 900 + 60 + 4. 
9000 + 700 + 80 + 3 = 9783. 

2. The numbers may also be written in vertical columns : 

78964 9000 

700 

70000 80 

8000 3 

900 

60 9783 

4 
15. Some Points in Multiplication. 

1. When the multiplicand consists of several figures and 
the multiplier of but one, the pupils are usually instructed 
to begin at the units' place of the multiplicand and to 
multiply each of its figures in succession by the multiplier ; 
but this procedure is looked upon by the pupils as arbi- 
trary. If, however, taking the following example, 456 X 8, 
the multiplicand be separated into 400, 50, and 6, and 
the pupils told to find the sum of the products of these 
numbers each multiplied by 8, and then be led to see 
that the usual method is an abbreviation of this, all thought 
of arbitrariness will disappear. 

2. A development of multiplication like the following 
throws light upon points freciuenlly not well understood : 



Suggestions for Teaching Numbers. 149 

456 X 234 = 400 X 200 + 400 X 30 + 400 X 4 + 50 
X 200 + 50 X 30 + 50 X 4 + 6 X 200 + 6 X 30 -h 6 



X 4 = 



400 X 200 — 80000 

400 X 30 = 12000 

400 X 4 = 1600 

50 X 200 = loooo 

50 X 30 = 1500 

50 X 4 = 200 

6 X 200 = 1200 

6 X 30 = 180 

6 X \ — 24 



456 X 234 = 106704 
3. The following will show why the first figure Oi the 
product must be written under its multiplier: 

456 X At — 1824 
456 X 234 = \ 456 X 30 = 1368/0 
466 X 200 = 912/00 



456 X 234 =106704 
4. Various ways in which the partial products may be 
written : 



(l) 456 


(2) 456 


(3) 456 


(4) 456 


234 


234 


234 


234 


1824 


912 


912 


24 


1368 


1824 


1368 


18 


912 


1368 


1824 


12 








20 


106704 


106704 


105404 


15 
10 

16 

12 

8 



106704 



150 Science and Art of Education. 

5. Multiplication is a short method of finding the sum of 
a number of repetitions of the same number. 

ADDITION. MULTIPLICATION. 

. 4567 .4567 

4567 4 

4567 , 

4567 18268 



18268 

16. Sofne Points in Division. 

1. In multiplication, the number to be repeated and that 
denoting the number of repetitions are given to find the 
sum ; in division, the sum and the repeated number are 
given to find the number of repetitions ; or the sum and the 
number denoting the repetitions may be given to find the 
repeated number. 

2. Division may be regarded as a short method of per- 
forming a number of subtractions with the same subtra- 
hend. 

SUBTRACTION. DIVISION. 

2193 243) 2193 (9 

243 (l) 2187 



1950 
243 


(2) 


1707 
243 


(3) 


1464 

243 


(4) 


I22I 

243 


(5) 



978 



Suggisiioris for leaching Numbers, 1 5 ^ 

978 

243 (6) 



735 

243 


(7) 


492 
243 


(8) 


249 

243 


(9) 



3. In multiplication two factors are given to find their 
product ; in division the product and one of the factors, to 
find the other. 

I7» Important Divisibilities of Numbers. 

1. A divisor of several numbers is a divisor of their sum 
and difference. 

2. A divisor of the sum of two numbers and one of the 
numbers divides the other. 

3. A divisor of a number is a divisor of any multiple 
of it. 

4. A number is not divisible by any number but its 
factors. 

5. Dividing one of the factors of a number divides the 
number. 

6. Multiplying the divisor divides the quotient. 

7. Dividing the divisor multiplies the quotient. 
18. Meanings of Division, 

Of the following indicated division, 8 -r- 4, three different 
cases may be assumed : i. How many 4's are in 8 ? 2. 
Four is what part of 8 ? 3. What is J of 8 ? Although the 
three answers are expressed by the figure 2, no two of them 
represent the same thing. 



152 Science and Art of Education. 

19. Long Division. 

1. There is no real difference between short division and 
so-called long division; both aim at the same end and attain 
it by the same method. In short division most of the work 
is done mentally, but in long division the larger divisor 
makes it necessary to write it. 

2. A few short-division problems solved by the long- 
division process forms one of the best introductions to long 
division. 

20. Greatest Common Measures. 

1. The G. C. M. of several numbers is the largest factor 
common to all of them. It may consist of a prime number 
or of the product of several prime numbers. 

2. When the numbers are small, either of the following 
methods may be employed to find the G. C. M : 

9 = 3X3 

27 = 3X3X3 
45 = 3 X 3 X 5 
81 =3X3X3X3 
3 X 3 = 9, G. C. M. 3 X 3 = 9, G. C. M. 

3. When the numbers are large the method by division 
must be employed. Example : 

1679)7981(4 
6716 

1265)1679(1 
1265 

414)1265(3 
1242 

~^) 414 (18 
23^ 
184 
184 

By the following reasoning the method by division may 
be proved. In the example given, since the smaller of the 



39 


27 


45 


81 


ZZ 


9 


15 


27 


I 


3 


5 


_9 



Suggestions for Teaching JVumbers. 153 

two numbers is not an exact divisor of the larger, it is 
not their G. C. M.; but since (17, i) the number sought 
cannot be greater than (1256) the difference of the two 
numbers, it may be this difference ; a trial, however, shows 
that it is not, that it is not an exact divisor of (1679) ^^""^ 
smaller of the two numbers. Continuing the same reason- 
ing, we find that the G. C. M. cannot be greater than (414) 
the difference between 1265 and 1679, ^"^ ^^ a trial shows 
that it is not this difference, it may be the difference between 
a multiple of this difference and 1265, and this is found to 
be correct. 

21. The Least Cof?ijnon Multiple. 

1. The L. C. M. of several numbers is the smallest num- 
ber that contains each of them as a factor. 

2. The following are the two methods of finding the 
L. C. M.; that by factoring being the more easily explained. 

10= 2 X 5 

18 = 2 X3X 3 

56 = 2 X 2 X 2 X7 

75^3X5X5 



2 X 2 X 2X3X3X5X 5X7 = 12600. 



10 


18 


56 


75 


5 


9 


28 


75 


I 


9 


28 


15 



28 



2X5X3X3 X28X5 = 12600. 

22. Fractions. 

Though fractions have received considerable attention in 
the preceding pages, a few more thoughts concerning them 
remain to be given. 

I. A so-called compound fraction is an indicated multi. 
plication of fractions and not a fraction. 



154 Science and Art of Education, 

2. What is called a complex fraction is usually an indi- 
cated division of a fraction by a fraction. 

3. Multiplying a fraction by a fraction is taking such a 
part of the multiplicand as the divisor is of the unit. 

4. Dividing by a fraction is taking the dividend as many 
times as the divisor is contained in the unit. 

Remark. — From 3 and 4 of the foregoing, we note that, 
generally speaking, multiplying by a fraction divides, and 
dividing multiplies. 

5. A fraction may be reduced to its lowest terms by fac- 
toring both of its terms and cancelling the' common factors ; 
or, when both terms are large, by dividing them by their 
G. C. D. 

6. The following exercises may be used to show that the 
value of a fraction is not changed by multiplying or divid- 
ing both of its terms by the same number. 

i = } = f = I = tV, etc.; i = \^^^=^^ = /„, etc.; 
i = f = l = A= A,etc.; i = ^ = -5-33 =^4_ = _5^, etc. 

Remark. — Exercises like the foregoing may also be used to 
show that fractions of unlike denominators may be reduced to 
the same denominator and added or subtracted. 

7. Reducing several fractions of different denominators 
to the same denominator, by multiplying each numerator by 
all the denohiinators except its own, and all the denomina- 
tors together for a new denominator, multiplies both terms 
of each fraction by the same number. 

Illustrative example : 

2 ■ 4 I 6^ 2X5X7 __ 4X3X7 _ 6X3X5 _ 70 84 90 

3 5 7 3X5X7 5X3X7 7X3X5 105 105 105* 

Instead of finding the numerators by the preceding 
method, they may be found by dividing the common 



Suggestions for Teaching Numbers. 155 

denominator by the denominator of each fraction and mul- 
tiplying the quotient by the numerator. 

8. The numerator of a fraction may be divided by an in- 
teger (whole number) by first multiplying both terms of the 
fraction by the integer. 

Multiplying the denominator of a fraction by an integer 
divides the fraction by the integer, because it increases the 
ratio of the numerator to the denominator by the multiplier. 

9. No reason can be assigned for the inversion of the 
divisor in the division of a fraction by a fraction, but its 
correctness can be shown by means of an analytic explana- 
tion or demonstration. For example, let it be required to 
divide \ by f . Explanation : One third is contained in i 
three times, and f (being twice as large) one half as often ; 
that is, the number of times f are contained in i ; but in \ 
they are contained one eighth as often, and in \ seven times 
as often as in \, Statement of analysis: fXi>^i>^T — 
f X-J. In this statement we see that the divisor has been 
inverted ; and since the same will invariably be the case, we 
must infer that inverting the divisor and then multiplying 
give the correct result. 

10. The correctness of the foregoing may also be shown 
by means of a diagram. 

23. Decimals. 

1. Decimals are parts of things, or units, in which the 
division is made according to the scale of tenths. Decimal 
expressions, though not decimals, will here for convenience 
be treated as decimals. 

2. Decimals, being derived from integers, should be de- 
veloped from them. This may be done by continuing the 
division as far below the decimal point as may be thought 
necessary. The following example will indicate some of the 
steps that may be taken to introduce the subject : 



156 Science and Art of Education. 



1000, 100, 10, I, ~, — , . 1000+ 1004- 10+ 1 -I 1 1-^ — . 

10 100 1000 10 100 1000 

jL — • JL _ . ^ _ ^ _ ^°° ^ _ ^° 
10 * ' 100 " ' 1000 ' 10 ~ 1000' 100 "~ 1000* 

. , 100 , 10 , I III 

.I-f-.OI-f .001 = .III. = . 

1000 100 1000 1000 

I.I.I 100 , 10 , I III , , 
= = =.I + .01. + .001 =.111. 

10 100 1000 1000 1000 1000 1000 

3. The pupils should be led to see or to discover (i) that 
the point is the distinguishing mark of decimals, and that it 
is placed before them to separate them and to distinguish 
them from integers ; (2) that as the value of integers de- 
creases from left to right and increases from right to left, so 
also does that of decimals ; (3) that the value of a decimal 
depends upon its distance from the point ; and (4) that for 
every place a decimal is moved to the left it is multiplied by 
10, and for every place it is moved to the right is is divided 
by 10. 

4. Practice should be given in writing and reading deci- 
mals until the pupils can readily do either. Decimals should 
be read as if they were common fractions, the name of 
the last place to the right given as the denominator. Ex- 
ample : In 0.00456, the last place to the right is that of 
hundred-thousandths, the fraction is therefore 456 hundred- 
thousandths. 

Remark. — Decimals may be written in four different ways : 
I. In words (three hundredths); 2. In the common fractional 
form (yfo) I 3- With the per- cent sign {yf) ; 4. In the usual form 
(.03). 

5. Place of Point in Multiplication. — The following will 
show how the rule for the location of the point may be de- 
rived. Taking .5 X .4 as an example, if we discard the 
points and multiply 5 by 4, we obtain 20 as the product ; 
but the multiplicand is not 5, but yV of it, hence the prod- 
uct is yV o^ 20, or 2.0 ; and for a similar reason, since the 



Suggestions for Teaching Numbers. 1 5 7 

divisor is not 4, but yV ^^ i^) ^^^ ^^st found product is not 
2.0, but yV ^^ it> o'* -20. An examination of the number of 
places in the product shows that it is equal to the sum of 
those in the factors. In the same manner may the rule be 
derived when the factors contain two or more decimals. 

6. Place of Point in -Division. — If in .45 -^ .5 = .9, we dis- 
card the points and perform the division, we obtain 9 as the 
quotient ; but since the dividend is not 45, but y^^ of it, 
the quotient must be y^ of 9, or .09. This is the quotient 
obtained by dividing by 5, ten times the divisor, and is 
therefore yV of the correct quotient, or .9, Here we observe 
that the number of decimal places in the quotient is that by 
which those in the dividend exceed those in the divisor. 
The same kind of reasoning will discover the rule when 
both terms contain several decimals. 

24. Analysts. 

I. Proportion. — The problems usually found under the 
head of proportion in books on arithmetic can, with more 
benefit to the pupils, be solved by analysis and cancellation. 
Simple problems, such as are found on page 128, should at 
first be given, and their length and difficulty increased as 
the pupils show themselves able to master them. 

Problem i. — If in 9 days, of 8 hours each, 20 men can 
build a wall 40 feet long, 2 feet thick, and 6 feet high, how 
many men would be required to build a similar wall 60 feet 
long, 3 feet thick, and 5 feet high, in 15 days, of 12 hours 
each ? 

Remark. — Before the analysis of a problem is commenced, a 
statement of the conditions of the problem should be made. It 
should also be observed that the first term of the analysis 
should be of the same kind as the one required. 



Statement of Conditions : 










M. d. 


hrs. 


ft. 1. 


ft. th. 


ft. h. 


20 9 


8 


40 


2 


6 


? 15 


I? 


60 


3 


5 



158 Science and Art of Education, 

Statement of Analysis : 

20Q181 iCoisit; 
-X^X-X-X X-X-X-X-X;,X^ = i5. 
I I 15 I 12 40 I 2 I 6 I 

Analysis, or Explanation. — Since the work can be done in 
9 days by 20 men, it would require 9 times as many men 
to it in I day, and -^ as many in 15 days as in i day. 
That is, if they work 8 hours a day ; but i hour a day would 
require 8 times as many men as 8 hours, and 12 hours a day 
yV as many as i hour. That is, if they make it 40 feet 
long; but i foot long would require :fV ^s many men 
as 40 feet, and 60 feet 60 times as many as i foot. That is, 
provided they make it 2 feet thick ; but i foot thick would 
require one half as many as 2 feet, and 3 feet three times as 
many as i foot. That is, if they make it 6 feet high ; but i 
foot high would require one sixth as many men as 6 feet> 
and 5 feet five times as many as i foot, t 

Of every problem twice as many problems can be made 
as it has terms. Of the following eleven, made from the 
foregoing, only the statements of the conditions and of the 
analyses will be given. The problems can be read from the 
statements. 

M. d. hrs. ft. 1. ft. th. ft. h. 



Problem 2. I5 I5 


12 60 3 5 


? 9 


8 40 2 6 


15xI5x-^X^-'x-^x5-° 
I I 9 I 8 I 


xi°x-'x^x^x? = 

I 3 I 5 I 


M. d. 


hrs. ft. 1. ft. th. ft. h. 


Problem 3. 20 9 


8 40 2 6 


15 ? 


12 60 3 5 


9x?5>xi-x-'xix-^^ 

I I 15 I 12 40 


x5?x5x?x-'x-5 = 

I 2 I 6 I 


M. d. 


hrs. ft. 1. ft. th. ft. h. 


Problem 4. I5 I5 


12 60 3 5 


20 ? 


8 40 2 6 


L^x'-^xi-x^^x^xi 

I I 20 I 8 60 


40 I 2 I 6 

X - X - X - X - X - = 
I 3 I 5 I 



15. 



Suggestions for Teaching Numbers, 159 







M. 


d. hrs. 


ft. 1. ft. th. ft. h. 


Problem 5. 




20 


9 8 


40 2 6 






15 


15 ? 


60 3 5 


I I 


X 


ix?x 
15 I 


- X - X - 
15^40^ I 


x^xfx^xf = i2. 






M. 


d. hrs. 


ft. 1. ft. th. ft. h. 


Problem 6. 




15 


15 12 


60 3 5 






20 


9 ? 


40 2 6 


ix'^ 


X 


1x1x5x^x1° 

20 I 9 60 I 


X-^X^X^X^= 8. 
3 I 5 I 






M. 


d. hrs. 


ft. 1. ft. th. ft. h. 


Problem 7. 




20 


9 8 


40 2 6 






15 


15 12 


? 3 5 



40 I 1^ \ 1^1122161 

-x-X-X-X-X5X-X-X-X-X-=6o. 

120191811315 

M. d. hrs. ft. 1. ft. th. ft. h. 

Problem 8. 15 15 12 60 3 5 

20 9 8 ? 2 6 

60 I 20 101831^1 
-X -X- y- X^X- X-X-X-X-X2 = 40. 
I 15 I ^ 15 I 12 I I 2 I 6 ^ 

M. d. hrs. ft. 1. ft. th. ft. h. 

Problem 9. 20 9 8 40 2 6 

15 15 12 60 ? 5 

?x-!-x^xixi^x^xi-^x^-2xix5xI= 3. 
I20I9I8I I 60 15 -^ 

M. d. hrs. ft. 1. ft. th. ft. h. 

Problem 10. 15 15 12 60 3 5 

20 9 8 40 ? 6 

^wi 20 I I 8 60 I ^i 
-X-X — X-X-X-X-X-X-X-X^= 2. 
I 15 I 15 I 12 I I 40 I 6 

M. d. hrs. ft. 1. ft. th. ft. h . 

Problem ii. 20 9 8 40 2 6 

15 15 12 60 3 ? 

5x-Lxyxix"ix-'x-'-'xl°x^x?x-'= 5. 

J 20 J 9 I 8 I I 60 I 3 ^ 



i6o Science and Art of Education. 

M, d. hrs. ft. 1. ft. th. ft. h. 

Problem 12. 15 15 12 6q 3 5 

20 9 8 40 2 ? 

5x-Lx2?x-'-x?x-'-x?x*"x-^'-x?xJ= 6. 

I 15 I 15 I 12 I I 40 I 2 

Remark. — Until the pupils can themselves see the reason for 
doing so, they may be told that the reasoning must begin and 
the units be taken in the horizontal line of the statement of the 
conditions in which all the numbers are given. As will be 
seen, the reasoning in all the foregoing begins in the upper 
line. 

The following problems can all be solved by the foregoing 
method, and in most cases with much less work than the usual 
method requires : 

I. If it takes 13,500 bricks, 8 inches long, 4 inches wide, 
and 2 inches thick, to build a wall 200 feet long, 20 feet 
high, and 16 inches (i^ feet) thick, how many bricks, 10 
inches long, 5 inches wide, and 2\ inches thick, would be 
required to build a wall 600 feet long, 24 feet high, and 20 
feet thick ? 

Brick. Wall. 



Bricks. 


in. 1. 


in. w. 


in. th. 


ft. 1. 


ft. h. 


in. th. 


13500 


8 


4 


2 


200 


20 


16 = f ft. 


? 


10 


5 


2i 


600 


24 


20 ft. 



13500 x|x3^^xfxixfxfxix^ioX^x^Vx¥xixfx-^-j^ = 
373>248. 
2. If 6 men in 4 months, working 26 days for a month 
and 12 hours a day, can set the type for 24 books of 300 
pages each, 60 lines to the page, 12 words to the line, and 
an average of 6 letters to the word, in how many months of 
24 days each, and 10 hours a day, can 8 men and 4 boys 
set the type for 10 books of 240 pages each, 52 lines to the 
page, 16 words to the line, and 8 letters to the word, 2 boys 
doing as much as a man ? 

M. mo. d. hrs. books, pages, lines, words, letters, 

(8men4-4boys 6 4 26 12 24 300 60 12 6 
= 10 men.) 10 ? 24 10 10 240 52 16 8 
fXfXfoX^XArX-V-XTVX^VX-V-XaAoX^I^X-cVxY-XrsXY-XiX 
1=1. 6+, 



Suggestions for Teaching Numbers. 1 6 1 

3. How many cords of wood in a pile 80 feet long, 12 
feet high, and 4 feet thick ? 

Cords. ft. 1. ft. h, ft. th. 

• I 8 4 4 

? 80 12 4 

i X i X ^«- X i X -V- X i X f = 30. 

4. How many cubic yards of earth is taken from a ditch 
120 feet long, 4 feet wide, and 9 feet deep ? 

Cu. yd. ft. 1. ft. w. ft. d. 

1333 

? 120 4 9 

i X i X H" X if- X i X f X i X f = 160. 

5. How many perches of stone in a wall T^d feet long, 2|- 
feet thick, and 5 feet high ? 

Perches. ft. I. ft. th. ft. h. 

I \(y\ \\ I 

? 36" 2^ 5 

+ X 3V X f X \«- X i X f X i X f X f = 18.18 +. 

6. What is the number of bushels of wheat a bin 9 feet 
long, 5-I feet wide, and if feet deep. 

Bu. ft. 1. ft. w. ft. d. 

(li cu. ft. = I I ij I I 

bu., nearly.) ? 9 53 3* 

i X i X t X f X i X \^ X i X -V- = 144. 

7. Required the number of gallons of water in a tank 12 
feet long, 3-^ feet wide, and if feet deep. 

Gal. ft. 1. ft. w. ft. d. 

(i cu. foot = i\ gal- 7^ III 

Ions, nearly.) ? 12 3;^ if 

-V- X Jf- X i X Y X i X Y- X iX f = 500, 



1 6 2 Science and Art of Education. 

II. Simple Interest. — Interest can more readily and 
intelligently be taught to beginners by stating the number 
of cents on the dollar than by using the term per cent. 
Six cents on the dollar, for example, is 6 per cent. 

If the time is given in years and months, it may be re- 
duced to the fraction of a year or to months ; and if it is 
given in years, months, and days, it is more conveniently re- 
duced to days, counting 30 days to the month and 12 
months to the year. If exact interest is required, 365 days 
must be taken for a year, and the correct number of days 
between the dates found. 

Problem i : What is the interest of $804 dollars, at 6c 
on the dollar, for 9 years 10 months and 5 days ? 

p. int. time. 

$1 6c I yr. 



? 9 yr, 10 mo. 5 d. 

|XAf^Xs^Xi\«-=:$475-°3- 

P. int. time. 

Proble^n 2 : $1 ? i yr. 

$804 47503c 9 yr- 10 mo. 5d. 

^f^ X ^i^^ X ^-P X 3^4 = 6c. 

p. int. time. 

Problem 3 : $1 6c i yr. 

47503c ? 



^f^ X i X ^H-^ X 8^4 =: 3545 days = 9 yr. 10 mo. 5d, 

P. int. time. 

Problem 4 : $1 6c ? 

$804 47503c 9 yr. 10 mo. 5d, 

'^¥-^ X ^iof X |X H^ == 36od. = I yr, 



Suggestions for Teaching Numbers. 1 6 



p. int. time. 

Problem 5 : $1 6c i yr. 

? 47503c 9 yr. 10 mo. 5 d. 

i X i X ^^^ X ^« X 3A5 = $804. 

p. int. time, 

? 6c I yr. 

Problem 6 : $804 47503c 9 yr. 10 mo. 5d. 

n^ X TTka X \X ^^^ X 7i<r =$1. 

The following problem contains dollars and cents in the 
principal and a fraction in the rate : 

p. int. time. 

Problem 7 : $1.00 5ic i yr. 

$345-75 ? 2 yr. 4 mo. 9d. 

3-842 
4 11525 283 

V X Tk X J-^"-^ X ^^ X H^ = $43.49. 

9 
3 

Explanation. — The interest of 100 cents, or $1, is ^- 
cents, of I cent it is yiij as much, and of 34575 cents it is 
34575 times as much as of i cent. That is for 360 days, or 
I year ; for i day it is -^\-^ as much, and for 849 days (2 yr. 
4 mo. 9 d.) 819 times as much as for i day. 

The cancellation can be considerably simplified by di- 
viding both numerator and denominator by 1000 ; that is, 
by striking out 3 noughts of the denominator and pointing 
off three places in the principal of the numerator. 

By continuing the division of the numerator by the de- 
nominator or its factors to as many decimals as may be nec- 
essary for accuracy, the whole denominator can generally 
be cancelled. 

III. Percentage. — The following, is one of the simplest 
methods of introducing percentage, the pupils having 
learned the numbers at least as far as loo. The tegcbef 



164 Science and Art of Education. 

should say that the terms here presented are only other 
names for the fractions with which they are already famil- 
iar. Instead of writing the words per cent., the sign (^) 
meaning hundredths should be used. 

100^ = 11^-1, the whole of anything ; 75^ = ^^^ = f ; 
5o^ = wo=i; 25,'^ = -^ = ^; 20^ = ^ = i; 10^ = 

I'A = TO ; 5^ = yfo = h ; i^ = T¥o ; 90^ = A ; 80^ = 

f; 7o^ = yV; 60^ = I; 40^^=1; -T^^li^l', 16^0 = i ; 
i2ifo = i; and 6i^ = yV 

Many oral problems should be given to familiarize the 
pupils with the new terms and with the applications of per- 
centage. 

The analytic method of solution applies as well to per- 
centage as to proportion and other similar subjects. 

Note. — Every number is 100 per cent (or the whole) of 
itself. 

Problem i : 4 is what ^ of 5 ? 

No. % ^ 

(No. = number.) 5 100 J-^o x ^ x | - 80. 

4 ? 

Explanation. — Since 5 is 100^, i is \ as many^, and 4, 4 
times as many as i. 

Frobkfn 2 : 4 is 80^ of what number ? 

No. % 

? 100 f X sV X H^ = 5. 

4 80 

Explanation. — Since 4 is 80^ of the number, i^ of it is 
-^ of 4, and 100^, or the whole of it, 100 times as many. 
Problefn 3 : What is 80^ of 5 ? 

No. % 

5 100 f XTh-X-V-=^4. 
? 80 

Problem 4 : If 4 is 80^ of some number, what^ of it is 5 ? 



Suggestions for leaching Numbers, 165 

No. % 

5 ? Y X i X f = 100. 

4 80 

Proble7n 5 : By selling my cow for $40 I gained 25^ on 
its cost ; what did it cost me ? 

Remark.— Adding 25^ (the gain) to 100^ (the cost) and 
we have 125 ^ of the cost. 

No. fo 

. 40 125 -v-x 14^x1^ = 32. 
? 100 

Problem 6 : On counting my money I found that $24 was 
20^ less than I had when I left home ; how much had I 
spent ? 

Remark. — Subtracting 20^ from 100,^, and we have 80^ ; 
what was left. 

No. % 

24 80 H- X sV X \«- = 6. 

? 20 

Remark. — The pupils should, as soon as possible, be led to 
solve percentage problems by the fractional method. 

IV, True Discount. — In true discount the present worth 
corresponds to the principal in simple interest, the dis- 
count to the interest, and the debt, or sum discounted, to 
the amount. When the sum to be discounted, the rate, and 
the time are given, to find the present worth, the problem is 
the same as having given the amount, rate, and time, to find 
the principal. 

Problem : I owe $540, due in 4 years without interest, 
money being worth $5^ ; what sum would discharge the 
debt to-day ? 

Explanatioji. — As $540 is the amount of the present 
worth of the debt due in 4 years at 5^0, we must find the 
amount of $1 for 4 years at 5 ic The interest of $1 at 5^ 
for 4 years is 20 cents, and this added to the dollar makes 



1 66 Science and Art of Education, 

$1.20, the amount. Now, considering it as a case in simple 
interest, and we have given two amounts and the principal 
of one, to find that of the other. As will be observed, 
therefore, two operations are required ; the first, to find the 
amount of $1 ; the second, to find the present worth. The 
following are the statements for the second part of the so- 
lution : 

p. amt. 

(p. — present $1.00 $1.20 

worth.) ? 540 

H- X lio X -^^1^ — 450 — required present worth. 

V. Bank Discount. — Bank discount differs from true 
discount in two respects : i. In being the interest on the 
face of the note ; 2. In adding 3 or 4 days of grace to the 
required time. 

Remark.— The face less the discount equals the proceeds. 

Explanation. — Since the proceeds are found by subtract- 
ing the interest (discount) from the face of the note, hence, 
when the proceeds are given and the face value is required, 
the proceeds of $1 must be found; then the proceeds of the 
dollar bear the same relation to the dollar as the given pro- 
ceeds do to the required face of the note. Here, also, it 
will be observed, two operations are required : the first, to 
find the proceeds of one dollar ; the second, to find the re- 
quired face of the note. 

p. int. time. 

I. $1.00 6c. 36od. fX^foX-V — .0155, discount on $1. 
$1.00 ? 93d. i.oo — .0155 — .9845, proceeds of $1. 

Proceeds. Face. 

$0.9845 $1 

$600 ? 

(0 + X r^-h-i X H^ = 609,446 + required face. 
(2) i X ^sVg X ^ii^^iiii ^ 609,446 + " 
Remark.— Number (2) is, perhaps, the simpler operation. 



Suggestions for Teaching Numbers. 167 

VL Time Problems. — Time relations are best repre- 
sented by the divisions of horizontal lines. 

Proble7n i : What time of day is it when the time to mid- 
night is twice the time past noon ? 

Solution. — N. ^- j M^ N. The time past noon and 

the time to midnight equal 3 times the time past noon ; 
hence, the time from noon to midnight, or 12 hours, is 3 
times the time past noon, and once the time past noon, or 
the required time, is 4 o'clock p. m. 

Problem 2 : What time of day is it when the time past 
noon is ^ of the time to midnight ? 

Solution. — ^-H fn , H^ -^- The time to midnight and 

half the time to midnight, or f the time to midnight, equal 
the time from noon to midnight, or 12 hours ; hence, ^ the 
time to midnight equals ;| of 12 hours, or 4 hours, and f of 
the time to midnight equal 8 hours, and the time is 4 
o'clock P.M., the same as that of the previous problem. 

Problem 3 : Required the time of day when the time 
past noon equals \ of the time past midnight. 

Solution.— M- N.^ , ^ , M^-^ P - Since the time past 
noon is \ of the time past midnight, the time before noon, 
or from midnight to noon, must be f of the time past mid- 
night, and this is 12 hours. If f of the time past midnight 
is 12 hours, \ of it, the time past noon, is ^ of 12 hours, or 
4 hours, and the required time is 4 o'clock p.m. 

Problem 4 : What is the hour of day when the time to 
noon is \ of the time to 2 o'clock p.m ? 

Solution,— t |-HN'H , H , H , H^ P.M.. If the time to 

noon is \ of the time to 2 o'clock p.m., then the time past 
noon, or 2 hours, must be -| of the time ; and if f of the 
time equal two hours, \ of it, or the time to noon, is 5 of 2 
hours, or \ an hour, and the time required is 11 o'clock a.m. 



t68 Science and Art of Education. 

VII. Age Problems. — Age relations are best repre- 
sented by the divisions of vertical lines. 

Remark. — The difference in the ages of two persons re- 
mains the same as long as both of them live. This fact serves 
as key to the solution of age problems. 

Problem i : John is now 4 years of age and Frank is 10 ; 
in how many years will John be f as old as Frank ? 

Solution. — The difference of their ages is 6 years, Frank 
and this, as the lines show, will be -g- of Frank's )4 
age at the required time. As John's age will then —^^" 
be f of Frank's, it will be twice 6 years, or 12 
years, and since he is now 4, the required time is 
the difference between 12 and 4, or 8 years. 

Problem 2 : Sarah is 11 years of age and Alice is 20; 
how long since Sarah was \ as old as Alice ? 

Mice 

Solution. — The difference oftheir ages, 9 years, 
was \ of the age of Alice at the required time ; 
hence Alice was 18 years of age when Sarah was 
\ as old ; and this was two years ago. ^ 

i 

Problem 3 : Twelve years ago I was \ as old as Mr 
Jones, now I am f as old ; what is my present age ? 

Solution. — Looking at the divisions of the Jones 

lines, and we find that 12 years ago the dif 
ference of their ages was \ of Jones' age, ^ 
and now is -^ of it ; but as the difference + % = -^ 
does not change, evidently \ of Jones' age y/^ j 
12 years ago was \ of what it now is, and f 
of it then f of what it now is. From this we see that 12 
years ago Jones was f as old as he now is, and since he now 
is f, 12 years must be the \ which he has in these years 
added. If 12 years equal \ of Jones' present age, he is 
now 36, and I am f of 36, or 24. 



_ ^Sarah 



-pl2 years 



"- 9 years ^' 



Suggestions for Teaching Numbers. 169 

Problem 4 : Sarah is now 3 times as old as Martha, but in 
9 years will be twice as old ; how old is each now ? 

Solution. — From the divisions of the lines Samh 

we see that the difference of their ages is 
now twice Martha's age and in 9 years will be 
once her age ; but as the differences are the 
same, twice Martha's age now is once what 
it will be in 9 years, and once her age now is 
\ of what it will then be. Since Martha's age Martha 

will then be f of what it now is, she must in 
9 years add the other \ ; and if 9 years equal \ of her age 
then, it must be her present age, and Sarah's is 3 times 9, 
or 27 years. 

Remark. — If in the foregoing problem the question were, 
how old will each be then, the following would be the solution, 
differing only in one point from the preceding : 

Solution. — From the divisions of the lines we observe 
that the difference of their ages is now twice Martha's age, 
and in 9 years will be once her age ; but as the differences 
are the same, 2 times Martha's age now is once what it will 
be in 9 years, and once her age now is ^ of what it will 
then be. Since Martha's age will then be f of what it now 
is, she must in 9 years add the other i ; and if 9 years 
equal \ of her age then, f, or the whole of it, will be 18 
years, and Sarah's will be 2 times 18 years, or 36 years. 

VIII. Watch and Chain Problems. — Problem i : A 
man had 2 watches and only i chain. If the chain be put 
upon the first watch it will make its value twice that of 
the second ; and if it be put upon the second watch it will 
make its value 3 times that of the first ; if the value of the 
first watch is $30, what is that of the second, and of the 
chain ? 

Remark. — The equations are designed to indicate the rela- 
tions of the parts of the problem and thus to aid the solution. 



I7<5 



Science and Art of Education. 



f. w. & ch. 



2 s. w.; 3 s. w. = whole. 

.*. s. w. = 1^ of whole. 

. w. & ch. = 3 f. w.; 4 f. w. = whole. 

.•. f. \v. — \^oi whole. 



Solution. — As the first watch 
and chain are together worth 
twice as much as the second, 
if to this we add the second 
we have 3 times the second for the whole, and consequently 
the second is \ of the whole. In the second condition we 
have the second watch and chain worth 3 times as much as 
the first ; if to this we add the first, we have 4 times the first 
for the whole, and the first is \ of the whole. Since the first 
watch is worth $30, the value of the whole is $120, that 
of the second $40, and -fi^, that of the chain, $50. 

IX. Fish Problems. — Problem : The head of a fish 
weighs 10 lbs., the tail weighs as much as the head and \ 
the body, and the body weighs as much as the head and 
the tail ; what is the weight of the fish ? 

Solution. — Make a sketch of the fish, divide it into head, 
body, and tail, and upon each part place the number given 
to it in the problem. 




The sketch of the body of the fish shows that 20 lbs. 
and \ the body is the whole body, but whatever added to a 
\ makes the whole must be the other half ; therefore 20 lbs. 
must be the other \. If \ the body weighs 20 lbs., the 
whole of it weighs 40 lbs., and the fish — head 10, body 
40, tail 30 = 80 lbs. 

X. Hound-hare Problems. — Problem i : A hare is 20 
leaps before a hound and takes 4 leaps to the hound's 3, 
but 3 of the hound's leaps are equal to 6 of the hare's ; how 
-many leaps must each take until the hare is caught ? 

Solution : First make a graphic representation or illustra- 
tion of the conditions of the problem. 



Suggestions for Teachuig Numbers. 



171 



hare. 
4 
6 



An examination of the accompanying repre- hound, 
sentation of the conditions of the problem 3 
shows that while the hound takes 6 of the •'' 
hare's leaps the hare takes but 4, and thus loses 2. If the 
hare loses 2 in running 4, to lose i it must run \ of 4, or 2, 
and to lose 20 if must run 20 times 2, or 40. 

When the hound runs 3 it gains the 2 which the hare 
loses; hence, to gain i it must run \ of 3, or i|-, and to gain 
20 it must run 20 times ij, or 30. 

Problem 2 : A rabbit is 60 leaps before a hound and 
takes 9 leaps to the hound's 3, but 2 of the hound's equal 7 
of the rabbit's ; how many leaps will each take until the 
rabbit is caught ? 

Solution. — Make a graphic representation of hound, rabbit. 

the conditions, and then, as the numbers of ^ ^ 

' ' 2 = 7 

the hound's leaps are unlike, multiply each of —^ 7 

the conditions by such a number as shall 6 = 21 
make them alike, or the same. The remainder of the solu- 
tion is the same as in the previous problem. 

XI. Alligation. — Problem : How shall I combine sugars 
that cost me 6c, 7c, 13c, and 14c a lb. so that I may be able 
to sell the mixture at 9c a lb ? 

Note. — The only conditions to be observed in the solution 
of problems of this kind are (i) that the sum of the gains shall 
equal that of the losses, and (2) that every ingredient upon 
which there is a gain shall be combined with one upon which 
there is a loss. 

Remark. — The arrangement of the solution of the problem 
here given is the usual one, except the horizontal line that 
separates the losses and gains. 

Solution I. — On a pound of 6c 
sugar sold for 9c there there will be a 
gain of 2)^, on a pound of 7c sugar 
sold for 9c there will be a gain of 2c, 
on a pound of 13c sugar sold for 9c 
there will be a loss of 4c, and on a 



p 


rices 


Diff. 


1 


:i 




6 


3 


5 


4 


erage 


7 


2 


2 


5 


9 




















13 


4 


1 


3 




U 


5 


3 


)i 



1 7 2 Science and Art of Education. 

pound of 14c sugar sold for 9c there will be a loss of 5c. 
Now, since the gains and losses must equal each other,* 
if we take 5 pounds of that on which there is a gain of 
3c and 3 pounds of that on which there is a loss of 
5c, the two will balance each other ; and if we take 2 
pounds of that on which there is a gain of 2c and i pound 
of that on which there is a loss of 4c, they will balance 
each other. 

Solution 2. — If we take 4 pounds of that on which there 
is a gain of 3c and 3 pounds of that on which there is a 
loss of 4c, they will balance each other ; and if we take 5 
pounds of that on which there is a gain of 2c and 2 
pounds of that on which there is a loss of 5c, they will bal- 
ance each other. 



pan % 

Geography. 

Introductory Consider atiofis. — Geography is generally 
classed among the dry subjects, but the dryrress is not so 
much in the subject as it is in the teachers and the teach- 
ing. A real teacher can invest any subject with interest, 
but a lesson-hearer kills even the interest that naturally in- 
heres in a subject. 

Much of the matter that has in the past been taught as 
regularly belonging to geography, and not a little of that 
still taught to children, is not of the most inviting nature. 

No greater mistake can be made in teaching beginners 
than requiring them to shut their eyes to the world in 
which they live and to look into a book where all is strange 
and meaningless ; yet this is the method generally pursued. 
The children's experiences, which should be used as start- 
ing-points, are not only ignored, but regarded as of no 
value. 

Definitions, of whose meaning the children can have no 
idea, are frequently the food which their mental stomachs 
are given to digest. After they are supposed to have 
learned these, then, with their undeveloped imagination, 
they are expected, from the study of a globe, to form a 
conception of the earth as a whole, and afterwards to di- 
vide it into its so-called members, and these again into sub- 
members, and so on, until the smallest division has been 
reached ; this method of procedure being followed, upon 
the ground that in everything studied or conceived, that is 
composed of parts, one must begin with analysis — with the 

173 



1 74 Science and Ai't of Educatioti. 

whole and go to its parts ; but such a strain at universality, 
the law "from the whole to its parts," will not bear ;, and a 
little reflection upon the manner in which the mind builds 
its spacial forms will show the erroneousness of the 
method. 

The suggestions which follow are a departure both in 
matter and method from what usually passes for geography, 
but are in harmony with the best in that line of instruc- 
tion. 

The study of geography should be commenced in the 
primary school, with what the children can observe, and 
should be largely conversational. Its object should be to 
awaken an interest on the part of the children in the 
boundless forms of nature that meet them on every hand. 

The work of the primary and others of the lower grades 
of schools should embrace the following topics : 

1. Land. — i. Its forms in meadows, uplands, plains, hills, 
and mountains. 

2. The material of which land is composed, namely, soils 
and rocks. 

3. The materials of which the different kinds of soils 

are composed, namely, loam, sand, gravel, and clay. 

Remark. — The children should examine the different kinds 
of soil. 

4. The part each of the two soils (top and sub) performs. 
in the growth of vegetation. 

5. The kinds of soil certain crops demand, the prepara- 
tion and fertilization of the ground for the reception of the 
seeds, the time and mode of planting, and care of the 
plants. 

6. The changes which land-forms are undergoing, and 
their causes. 

7. The influence of land-forms upon climate. 

Note. — The children should make models in sand of the 
forms of land which they have observed ; this will enable them 



Suggestiofts for Teaching Numbers. 175 

later on in their study, when observing a model, to look with 
their imagination beyond it to that which it represents. 

2. Water. — i. The forms of water — ice, liquid, and 
vapor. 

2. How ice and vapor are formed and how returned to 
the liquid state. 

Note. — Whenever necessary and practicable, processes 
should be shown by means of experiments. 

3. Uses of ice and vapor in nature and to man. 

4. The formation of clouds, and their kinds. 

5. How rain is produced ; its uses or benefits, especially 
in the growth of plants. 

6. How springs, rivulets, creeks, rivers, ponds, and lakes 
are formed. 

Remark. — If no pictures are at hand that illustrate it, draw- 
ings can be made upon the blackboard to answer the purpose. 

7. The uses of water for drinking, washing, cooking, 
highways, as a force, and as a cooler and moistener of the 
air. 

3. Air. — I. The properties of the air. 

2. Its use in supporting animal and vegetable life. 

3. How winds are caused, and their kinds. 

4. The uses of winds in carrying vapor, purifying and 
cooling the air, propelling vessels, and turning machinery. 

5. Changes of temperature, how caused, and how meas- 
ured. 

6. Effects of changes of temperature upon animal and 
vegetable life. 

7. Protection of animal and vegetable life against great 
changes — summer's heat and winter's cold. 

4. Heat. — I. Its source or modes of production. 

2. Its effect upon different substances, 

3. Its uses. 



lyO Science and Art of Education. 

5. Plants. — I. Common varieties. 

2. The roots, and the part they perform in the growth of 
plants. 

3. The kinds of roots, also duration. 

4. Roots used for food. 

5. The- kinds and use of stem. 

6. Leaves, their forms and use. 

7. Flowers, their forms, use, and beauty. 

Remark. — The children should be trained to draw and 

paint plants, including flowers. 

8. The kinds of fruit or seeds, when they ripen, and 
their uses. 

9. Buds, what they contain, when they begin to swell, 
and why they do so. 

10. When the blossoms appear. 

11. When the fruit ripens. 

12. When the leaves begin to fall and what causes them 
to fall. 

13. How roots are protected from the cold in the winter. 

14. Food of plants, also cultivation. 

15. Medicinal and food plants, also plants that are poi- 
sonous. 

6. Domestic Animals. — i. Varieties, also form or struc- 
ture. 

2. Adaptation of structure to mode of life and subsist- 
ence. 

3. Grass, grain, and flesh eating animals, and habits and 
use of each. 

4. How fed, sheltered, and cared for. 

7. Wild Animals. — i. Varieties of form or structure. 

2. Adaptation of structure to mode of subsistence and 
defence or protection. 

3. Their habitat, or homes. 

4. Grass, grain, r4Ut, and flesh eating quadrupeds. 



Suggestions for Teaching Numbers. 177 

{a) Those that are useful for food, skins, and furs, 
{b) How captured, also how tamed. 

5. Flesh and grain eating birds, and varieties of each. 
{a) How they fly, and how they hold themselves to limbs 

of trees and other objects. 

(^) How and when they secure their food. 
[c] How they are captured ; also how tamed. 
{a) Which used as food, and why. 

6. Fish, their varieties, also adaptation of structure to 
medium in which they live. 

(a) How they move themselves, and adaptation of form 
to mode of movement. 

{B) Their food, and how they secure it. 

{c) Which are used for food. 

{d) Methods of catching, preserving, and preparing for 
the table. 

Among animals should also be included insects and rep- 
tiles, the latter embracing lizards, turtles, tortoises, frogs, 
toads, and serpents. An examination and study of the 
nature and habits of these will create an interest in them 
in the children and will lead them to see that none of 
God's creatures are useless, but that all of them when prop- 
erly understood have their purpose, and instead of being 
our enemies are our friends. 

Remark. — Microscopes are necessary for some of the vvork 
referred to. 

8. Man. — I. His superiority to other orders of creation. 

2. House and home life in comparison with those of 
other living beings. 

3. Adaptation to a variety of occupations or employ- 
ments. 

4. His genius in using the forces of nature in the per- 
formance of labor and in surmounting obstacles. 

5. Modes of communication and travel. 



178 Science and Art of Education. 

6. Possibility and means of improvement. 

7. Means or modes of enjoyment. 

8. Pictures of the various races of the human family 
should be shown, and the race characteristics described. 

9. The effect of climate upon character and disposition 
should be explained. 

Remark. — As before stated, the foregoing work should, as 
far as possible, be informal and conversational. 

9. Study fr 0771 Maps and Models. — i. After the locality or 
community in which the children have their homes has 
been explored and studied as carefully and thoroughly as 
their age will permit, and a sufficient amount of experiences 
or apperceiving concepts stored away as constructive ma- 
terial, the children are prepared to extend, their vision, by 
means of the imagination, to the unseen. This they must 
learn to do by the use of pictures, maps, and models, and 
the best and most natural way to learn to understand a map 
or a model is to help to make one. The school-room is the 
most convenient and suitable thing to begin with in map- 
ping. 

2. To enable the children to judge of distances and 
areas they should have practice in making measurements. 
A yard, rod, or mile, either linear or square, should convey 
something definite to them. 

3. The cardinal points — east, west, north, and south- 
should be determined and marked or fixed upon the floor 
or elsewhere by means of lines connecting the opposite 
ones. 

4. A map of the school-room may be made upon the 
floor, but better upon a piece of heavy paper, say a yard 
square, or larger, if necessary, painted with a mixture of 
shell-lac (dissolved in alcohol) and lampblack. 

5. To make the map, the paper should be laid at some 
convenient place upon the floor. If the map is to be pro- 



Suggestions for Teaching Numbers, 179 

portioned to the size of the room — about an inch to a foot 
— the pupils should make the measurements, determine the 
proportions, and otherwise assist the teacher in the work. 

Frequent questions should be asked of the pupils with 
regard to the direction of lines, the location of seats, 
teacher's desk, and other objects. 

6. After the map has been made and the pupils can read 
it — name any object upon it pointed to by the teacher or 
one of their own number, also its direction from some point 
named — it may be hung upon the wall and again read. 
The latter reading will prevent the erroneous notion some- 
times formed by pupils that north is in a vertical line above 
south, or at the zenith. 

7. After the children have become familiar with the map 
of the school-room, the surroundings of the school, and, if 
in a town or city, some of the principal streets and build- 
ings may be added. Next, if it contains enough objects of 
importance, the county may be drawn, then the state, and 
other states separately and in sections or groups, until the 
whole country has been built or mapped, and studied, and 
the children can in imagination see it or any part of it. 

8. As will be noticed, the method here indicated is pro- 
gressive ; and to show more definitely how it may be carried 
out, the following suggestions are added : (a) If the school 
is in Pennsylvania, begin with that State ; and if the pupils 
have had little or no practice in drawing from memory, the 
teacher should make the first drawing, explaining his work 
as he proceeds. 

{h) For the first lesson, the pupils should prepare to draw 
the outline (of the state) and the rivers, and at the recita- 
tion make a sketch of their lesson upon the blackboard, 
and describe it, the teacher questioning them upon it. 

{c) For the second lesson, they should add to the first 
the mountains, prmcipal towns, cities, and other objects of 
importance. After they have completed their sketches or 



1 So Science and Art of Education. 

maps they should describe them, and the teacher should 
ask questions about the proportions, the relative position of 
objects represented upon them, etc. 

(^) For the third lesson, the second should be repro- 
duced as a review, and the outline and the rivers of New 
Jersey added. 

Remark i. — While the pupils are learning the maps, the 
teacher should, with all the helps at his command, such as 
models, pictures, and descriptions, enable them, in imagination, 
to see the states or countries of which the maps are represen- 
tations. 

2. — Descriptions, questions, and reviews of as many previous 
lessons as may be necessary should constitute a part of every 
recitation, 

{e) The fourth lesson should review the third and add to 
it the mountains, cities, etc., of New Jersey. 

(/) The fifth lesson should include the fourth and add 
to it Delaware, with all in it of importance. 

{g) For the sixth lesson, add to the iifth the outlines of 
Maryland. 

{h) For the seventh lesson, add to the sixth whatever 
may be considered of importance in Maryland. 

Remark. — The daily reviews should be spirited ; slow, sleepy 
work should not be permitted. The pupils should be trained 
to accurate rapid sketching or mapping. 

(/) If by this time the pupils have Pennsylvania well 
pictured in their minds, so that they can make a rapid and 
sufficiently accurate sketch of it and describe it, they may 
drop that state for a while and add to the others already 
drawn Virginia, then West Virginia, and so on, always, in 
the sketching, dropping those first made as soon as they are 
well fixed in the mind. 

(y) A daily review, either oral or by sketching, is a neces- 
sity, in order to connect the work from the beginning into 
a mental picture of the whole and to impress it firmly upon 
the mind, Frequently all the states that have been studied 



Suggestions for Teaching Numbers, iSi 

should be sketched as a whole. It is of far more impor- 
tance that what has been learned should find a permanent 
lodgment in the mind, than that more should be added to 
what is already fading. 

{Jz) After New Jersey has been added to Pennsylvania, 
instead of taking Delaware next, New York may be taken, 
then Connecticut, Rhode Island, Massachusetts, Vermont, 
New Hampshire, and Maine. 

Remark. — If the pupils can do so, they may group two, 
three, or more states together as a lesson. 

(/) The teacher should begin with the state in which his 
school is located and build from that out. He may, of 
course, begin with some other state, but it is more natural 
to begin at home. 

(;//) Only things that are important should be found in 
the sketches. 

(;/) Maps in which no exactness is required should be 
proportioned by the eye, without construction-lines. How- 
ever, if a pupil finds it difficult to give the desired shape to 
his map while preparing his lesson for a blackboard sketch, 
dividing his paper into rectangles, proportioned as nearly as 
possible to those made by the parallels and meridians of 
the maps in his book, will give him all the points he needs 
for the required form. 

(o) While the pupils are doing the v/ork here suggested 
they should also read the descriptive matter found in their 
books pertaining to the states which they are sketching, 
and should be questioned upon it by the teacher. What- 
ever of this matter may be considered of sufficient impor- 
tance to constitute a permanent possessipn of the mind 
should also enter into the daily reviews. 

(/) Climate and productions are not limited by political 
divisions, but belong to physical sections or regions, and, 
unless peculiar to a state, should be taught with the regions 
to which they belong. 



t&2 Science and Art of JEducatioH. 

{q) Definitions, instead of being the first thing presented 
to a learner, should generally be the last, and instead of 
being memorized from a book should as far as possible be 
drawn from examples or given from a knowledge of the 
subject. 

(r) To complete the map of North America, add to the 
United States the British possessions, Alaska, and Green- 
land, on the north, and Mexico, Central America, and the 
West Indies, on the south, grouping all into one picture. 

Remark. — Countries of which our knowledge is limited and, 
at best, inaccurate, should have only the outline, principal di- 
visions, rivers, mountains, and cities represented on the map. 

{s) South America may follow North America, then 
Africa, Europe, Asia, with its surrounding islands, Australia, 
and the more important islands of the Pacific Ocean. 

9. After a whole country has been mapped and impressed 
upon the minds of ^the pupils, its prominent or controlling 
physical features — those upon which the climate, produc- 
tions, etc., depend— should be studied on a relief map or 
on a model in sand. 

10. The study of the influence of the physical features of 
a country should be followed by the position of the country 
upon the globe, also its position with reference to other 
countries; and its intellectual, moral, and commercial posi- 
tion or standing among the countries of the earth. 

11. Countries should also be compared with each other, — 
their resemblances and contrasts noted, — as indicated in 
Part II, under Association. 

12. When each country has its place assigned upon the 
globe, and a picture of the whole has been formed in the 
minds of the pupils, then they are prepared to study it in- 
telligently as a whole, with its lands, waters, motions, forces 
operative upon it, diversities, and possibilities of life, etc. 
This is a study of cause and effect, and suitable only for 



Suggestions for Teaching Numbers. 1S3 

pupils of sufficient age and mental development to make 
broad generalizations. 

10. The Sa7id-box. — i. Elevations are best represented by 
means of models of sand, clay, putty, or paper pulp. A 
box of inch pine-boards, 3 ft. X 4 ft. X 3 in., painted on 
the inside with two or three thick coats of lead paint, 
placed upon trestles 2 ft. in height and containing about a 
peck or ten quarts of moulders' sand obtained at a foundry, 
is not expensive, and should be found in every school in 
which geography is taught. 

2. That the sand may at all times be ready for use, it 
should, when not needed, be kept in a rounded heap, 
pounded together to hold the moisture, and once a day, or 
oftener, sprinkled with as much water with a sprinkling 
can as will soak in without running off. 

11. Relief -7naps. — i. Instead of purchasing expensive 
relief-maps, every teacher can, with the assistance of his 
pupils, make his own maps of paper pulp. The pulp may 
be made in the following manner : Take white waste paper, 
(other paper may be used) and tear it into pieces about an 
inch square, until enough has been prepared to fill a com- 
mon-sized wooden water-pail or bucket. Pour enough hot 
water upon the paper to cover it two or three inches deep, 
and let it remain on it six or eight hours, or overnight. When 
ready to make the pulp, pour nearly all the water off, leav- 
ing only as much as may be necessary for the proper moist- 
ure of the pulp. Pour a quart or more (if the pail is nearly 
full of paper) of flour starch upon the paper, and work or 
mix it well in with the hands. Now, after pouring half the 
mass into another vessel, — the reduction being more easily 
and quickly made by taking a half-pailful at a time, — let 
each of three boys take in each hand a stick of hard wood 
about three feet long, three-fourths of an inch thick, and 
pointed at one end, and, sitting around the pail, drive them 
down through the mass as rapidly as possible (frequently 



Science and Art of Education. 



stirring the paper up from the bottom), until the desired 
condition of the pulp has been attained. Treat the other 
half in the same manner, and when the pulp has all been 
made, put it into a stone jar or crock, and cover it well, in 
order that it may hold the moisture until it is wanted to 
make the maps. 

2. To make the maps, take a piece of cardboard about 
an eighth of an inch in thickness, proportion the map to the 
one in the book used as a copy, making it two, two and a 
half, three, or four times the size of the copy. Cut the card 
at least two inches larger each way than the proposed map, 
to allow for an inch border all around. Next, draw the 
map upon the card, and when done, with the fingers put 
the pulp on, not thicker than an eighth of an inch, pressing 
it down well to make it adhere to the card. It is best to 
begin putting on the pulp around the outline, and after- 
wards to cover the remainder. No pulp should be put upon 
places intended to represent lakes, nor should rivers be 
covered with it. Simply press the pulp down on both sides 
against the river line, but not together, and when it dries it 
will sepatate along the line and represent the river. 

3. The maps can be made harder and more durable if, 
when dry, they are given a coat of gum-arabic. If desired, 
when the gum-arabic is dry the maps may be painted. 

4. To make the pulp adhere more firmly to the card- 
board, the latter should also be given a coat of mucilage 
inside of the outline. If this is done the mucilage should 
be allowed to dry before the pulp is put on. 



History. 

1. History is a record of human deeds, either of individ- 
uals or communities ; and since deeds, measured by moral 
standards, may be good or bad, the study of history is the 
study of conduct, the study of morals. Viewed in this 
light it has an important bearing upon the formation or 
growth of character. 

2. The greater the age a country has attained the larger 
its accumulation of historical matter at the command of the 
teacher. In wealth of material, it is true, the lands beyond 
the ocean surpass us ; but for the lessons that young 
Americans need to learn, our stores are not only ample, but 
superior to all others. Our soil has been consecrated to 
liberty. The heritage which our forefathers have left us is 
priceless. No other country upon the globe offers such 
opportunities for persons of ambition and worth to rise 
from the humblest walks in life to the highest. No law 
says, ^' thus far shalt thou go but no further." 

3. But those who are soon to become actors in the drama 
of life must not only be taught to appreciate the value of 
the freedom handed down to them, but, above all, how to 
maintain it ; they must be taught that "righteousness " in 
the individuals "exalts a nation," and that sin is a re- 
proach to any people. That character or worth makes 
the man must therefore be impressed upon the minds of 
the children so as to become the ruling principle of their 
conduct and lives ; and, to accomplish this end, no other 
branch of study furnishes material equal to that of history. 

185 



1 86 Science and Art of Education. 

4. Erroneous notions concerning the influence of certain 
kinds of matter prevail among teachers. For example, 
some labor under the delusion that the study of battles and 
bloodshed cultivates a spirit of patriotism and bravery. 
They do not distinguish between bloodthirstiness, and 
patriotism and bravery. Pupils nourished with the former 
diet — soon eagerly devoured by boys — frequently find the 
ordinary quiet walks of life too uneventful for " the fires 
that in them burn," and start on a career of lawlessness. 
The matter for class use for children should therefore be 
selected with care. 

5. Children should be taught, and early, too, that might 
does not make right in the eyes of the civilized world ; and 
that, consequently, decisions rendered by the sword — a 
relic of barbarous ages — are not to be relied upon as 
founded upon justice, nor as compatible with Christian 
civilization. 

6. We look to the past also for guidance in the future. 
For this purpose, however, much of the past is not only un- 
necessary, but useless ; hence teachers of history should 
exercise more judgment than is usually done in selecting 
that which has a direct bearing upon the points they are 
endeavoring to have brought out or discussed. When the 
pupils have reached a sufficient age and stage of mental 
development to do so, the selection of the matter should, as 
far as possible, be left to the exercise of their own judg- 
ment, the teacher simply stating the question to be dis- 
cussed. 

Remark. — A teacher of history should be free from all taint 
of political bias. A politician or partisan would on all occa- 
sions that offered themselves try to influence his pupils to 
become of his own sort, and thus defeat the object of the study. 

7. When not confined to the dry bo:.ies of the subject, 
such as uneventful administrations and the like, or to the 
packing of the memory with dates and less important facts, 



Suggestions for Teaching Numbers. 187 

and when taught inteHigently, history is a study of infer- 
ence, induction, one of the best thought, or logical, subjects, 
and should not fail to be intensely interesting. 

8. A teacher of history should be a good story-teller. 
He should be able to put dry facts into the form of stories 
and invest them with interest. Children are fond of stories, 
hence this is the best way of presenting the subject to them. 

9. If the formation of Christian characters is uppermost 
in the teacher's mind and his pupils are primarians, he can 
do nothing better than begin with Bible stories, those of 
the Old Testament, of Abraham, Isaac, Jacob, Joseph, 
Moses, Joshua, Samuel, Solomon, David, Elisha, Job, 
Isaiah, Jeremiah, Daniel, and, in the New Testament, 
Christ, whose life and teachings furnish endless material 
for building character. 

10. Following Bible stories may come the history of our 
own country, beginning with that of the community — town- 
ship or county — in which the school is located, bringing in 
the Indians, then the discovery and settlement of the 
country by Europeans, the encroachment of the white man 
upon the hunting-grounds of the Indians, the resulting 
animosities and strifes, etc. Special stress should be laid 
upon the events and influences that have controlled our 
growth and strength as a nation. 

11. With whatever matter the teacher may begin, whether 
Bible stories or the community, it should be given in the 
form of anecdotes and stories. No books should be used ; 
for there is no surer way to destroy interest in the subject 
and create a dislike for it than requiring the contents of 
books to be recited. However, when pupils are old enough 
to read understandingly they should be encouraged to con- 
sult books, and if the instructions they have received from 
the lips of the teacher have created the proper interest, 
they will gladly avail themselves of every opportunity to 
add to their stock of knowledge. 



I S8 Science and Art of Education. 

12. The matter for advanced pupils should frequently, if 
not generally, be given in the form of problems, or subjects 
for discussion, to give them practice in solving the perplex- 
ing problems that will meet them later in life and which 
confront the citizens of a country like ours. 

13. Teachers of history should bear in mind that a good 
knowledge of their subject of instruction does not neces- 
sarily imply a large accumulation of facts, but the ability 
to use those at command to the best advantage. 

14. A teacher of history should have all the appliances 
in the way of maps, charts, pictures, etc., that are neces- 
sary to give his pupils accurate and clear mental pictures 
of the events that present themselves in their lessons. It 
is more frequently on account of failures in imagining than 
of treachery of memory that lessons are unsatisfactorily 
prepared. The more real the teacher succeeds in making 
his instructions the better they will be comprehended and 
remembered. 

15. In history fully as much as in anything else, if not 
more so, daily reviews constitute a necessity. This is the 
only way to connect the work and to give it a permanent 
place in the mind. 



The Human Body. 

1. Of all the studies pursued in the schools, that relating 
to the knowledge and care of our bodies is least under- 
stood. It is perfectly safe to say that we do not under- 
stand how to live well, and the little we pretend to know we 
disregard. 

2. Our food is selected in almost total indifference of 
what the system craves in kind, proportion, and quantity. 
Some kinds come too frequently, others the reverse. Again, 
some that the system does not need, cannot use, and that 
therefore do harm, are provided, while others that are needed 
are not supplied. Neither the work performed nor the 
season is consulted. The same kind is frequently provided 
for all, whether suitable or unsuitable, whether their stom- 
achs can bear it or not. All are supposed to be alike. 

3. Cooking is done by persons who are ignorant of the 
laws of health ; frequently they do not know that there are 
such laws ; and, in the preparation, either destroy the most 
nutritious part of the food or permit it to evaporate. 

4. The amount of exercise needed is as little understood 
as the food required. Either too much is taken or not 
enough. It is scarcely known, and still less believed, that 
the human machine, like others not human, may be run too 
fast or too slow ; neither of which can long be done with 
impunity. The avenger invariably comes to call a halt. 

5. Rest, in the form of sleep, seems to be too much re- 

189 



1 90 Science afid Art of Education. 

garded as a thing that can be put off from time to time 
until there is nothing else to do ; but rest is fully as impor- 
tant as food, and while some may take so much as to be- 
come tired of it, others, and many of them, do not take 
enough, especially those who sleep by the clock and not by 
what they need. Fortunately, no one needs a clock to tell 
him when he has had enough ; he has his guage within him. 
Those who claim, as some do, that time does not permit 
them to sleep until they feel that they have had sufficient, 
earlier or later pay the penalty for their indiscretion or 
imprudence. 

6. Cleanliness of person is also a thing that needs more 
attention than it usually receives. Bathing, some think, 
should be done now and then, say once a week or month, 
just as there may be time to spare from other duties. But 
daily bathing, and during warm weather more frequent, is 
a necessity to comfort and health. The pores of the skin 
need to be kept open and active. 

7. Ventilation is seldom overdone, but in ninety-nine 
cases in a hundred the reverse. Pure air, if a little cool, is 
considered dangerous, while the rankest poison thrown off 
from the lungs and skin is, with the utmost composure and 
ignorance, time and again returned to the lungs, until 
almost complete stupor ensues. 

8. What are our schools doing to remedy these defects ? 
What are they doing to teach right living — well living ? 
Will memorizing the names and number of the bones, teeth, 
muscles, nerves, etc., usher in the wished-for millennium ? 
If it is supposed to do so, it has hitherto proved a signal 
failure ; and as the past has begn so will the future be, 
unless a wiser course be pursued. 

9. It is not unfair to ask, how many persons who under- 
take to give instruction in this all-important subject are 
competent to do so ? Not a few of them, judging from 
their frecjuent ailments^ do not know their own bodies wel! 



Suggestions for Teaching Numbers. 191 

enough to take care of them, yet do not hesitate to instruct 
others how to take care of theirs. 

10. The structure of the body is well enough understood 
to be taught with success. All that teachers need is to 
make themselves thoroughly acquainted with it. 

11. With regard to hygiene the case is different. All that 
can be learned from books and teachers in the present state 
of knowledge of the subject is some very general facts, 
nothing more. Until we shall have something much more 
definite than we now possess, and perhaps ever after, every 
one must study himself to learn what his well-being de- 
mands. This study must be inductive, and must deter- 
mine what, under varying conditions, is healthful and 
what injurious. 

12. Primary pupils should, as far as possible, be taught 
objectively, without the use of books. One of the best 
teachers' helps for giving such instruction is " Practical 
Work in the School-room," Part I — The Human Body, 
published by A. Lovell & Co., New York. Lessons on The 
Boy, in " Systematic Science Teaching," a recent work, 
published by D. Appleton & Co., New York, will be found 
exceedingly helpful in showing how such and all other 
science instruction should be given. 

13. For work above the primary classes the following or- 
der of presenting the subject may be followed : i. The 
framework or bones ; 2. The muscles ; 3. The skin ; 4. Di- 
gestion ; 5. Food ; 6. Circulation ; 7. Respiration ; 8. The 
nervous system ; 9. The senses. 

Remark. — In connection with each part or organ should 
also be taught, as far as possible or practicable, its care. 

14. All unimportant and unnecessary details and scientific 
terms — a burden to the memory — should be omitted. It is 
more important that pupils should become interested in the 
study of their own bodies than that they should know all 



192 Science and Art of Education. 



about it. The order of presentation should generally be, 
first the thing, then its name. 

15. To teach the human body successfully, either a skel- 
eton and models of the various organs are required or a 
manikin ; without either of these it is impossible to give 
the pupils an accurate conception of the forms and position 
of the internal organs. 



Civil Government. 

1. There is no reason why the children in the lower 
grades of schools should not be made acquainted with the 
elements of civil government. There is nothing difficult to 
understand about the subject, and if presented in an intelli- 
gent manner it is an interesting one. 

2. The first lessons should be about their own community, 
whether township or borough, and should include the fol- 
lowing : I. What public duties arc, and why public rather 
than private ; 2. By what officers the duties are performed; 
3. Whether officers are elected or appointed, and by whom, 
how, when, and for what length of term ; 4. How officers 
are installed, and when ; 5. How and by whom paid ; 6. 
Duties each is required to perform ; 7. Rights and duties 
of citizens ; 8. How rights are secured and wrongs re- 
dressed ; 9. What laws are, by whom made, for whose ben- 
efit, and why needed ; 10. What the laws forbid. 

Next should come the government of the county, and 
then as much of that of the State as the pupils are old 
enough to understand and to be interested in. 

3. No subject should be presented to pupils before they 
have reached a period of life at which they can be inter- 
ested in it ; hence the study of the general government 
should be left for the high-school. 

193 



part ^911^. 

Drawing. 

Drawing is a mode of expression that manifests itself 
early in children. Give one a pencil or a piece of crayon 
and he will try to express something, however crude or un- 
intelligible the performance may be. Since the desire or 
instinct exists, why not foster it ? What the children need 
at this period is encouragement, help. Hence, whenever 
anything presents itself in their lessons that admits of rep- 
resentation by lines or colors, let them try to draw it or 
paint, it ; and, when necessary, show them how to improve 
their work. If this course be pursued, drawing and paint- 
ing will be taught in the most natural way, and without a 
special class or period for it. 

195 



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all teachers the pleasure arising from following our 
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KELLOQQ'S ELEMENTARY PSYCHOLOGY. 

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ROOFER'S APPERCEPTION, 

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is published weekly at $2.50 a year and is in its 23rd >ear. 
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Keliogg's School Management. - - - - cl. 

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Patridge's Qumcy Methods, illustrated, - - cl. 

Quick's How to Train the Memory, - - - paper 

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♦Keinhart'8 Principles of Education, - - cl, 

♦ " Civics m Education, - - « - cl. 

♦Rooper's Object Teaching, - - - - cl. 

Sidgwick's Stimulus in School, - - - - paper 

Shaw and Donneli's School Devices, - - - cl. 

Southwick's Quiz Manual of Teaching, - - cl. 

Yonge's Practical Work in School, - « - paper 

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Augsburg's Easy Drawings for Geog. Class, - paper 

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♦Burnz Step by Step Primer, - - - » 

Calkins' How to Teach Phonics, - - - cl. 

Dewey's How to Teach Manners, - - - cl. 

Gladstone's Object Teaching, - - - - paper 

Hughes' How to Keep Order, - - - - paper 

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♦Keliogg's How to Write Compositions - - paper 

Keliogg's Geography by Map Drawmg - - cl. .50 
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Seeley's Grube Method of Teaching Aiitbmetic, cl. 1.00 

'* Grube Idea in Teaching Arithmetic - cl. .30 
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WoodhuU's Easy Experiments m Science, - cl. ,50 



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PRIMAEY AND KINDERGAETEN 

Calkins' How to Teach Phonics, - - - cl, 

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Autobiography of Proebel, - - - - cl. 

Hofifman's Kindergarten Gifts, - • - - paper 

Johnson's Education by Doing, - - - - cl. 

♦Xilburn's Manual of Elementary Teaching - 

Parker's Talks on Teaching, - _ - - 

Patridtre's Quincy Methods, - - - _ 
Rooper's Object Teaching, - - - - 
Seeley's Grube Method of Teaching Arithmetic, 

" Grube Idea in Primary Arithmetic, - 

♦Sinclair's First Years at School, - - : - 



cl, 
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1.20 

1.00 

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skSD all okbkk^ i6 

E. L. KELLOGG & CO., NEW YORK & CHICAGO. 



Aliens Mind Studies for Young Teac h- 

ERS. By Jerome Allen, Ph.D., Associate Editor of the 
School Journal, Prof, of Pedagogy, Univ. of City of 
N. Y. 16mo, large, clear type, 138 pp. Cloth, 50 cents ; to 
teachers, 40 cents ; by mail, 5 cents extra. 

There are many teachers who 
Know little about psychology, 
and who desire to be better in- 
formed concerning its princi- 
ples, especially its relation to the 
work of teaching. For the aid 
of such, this book has been pre- 
pared. But it is not a psychol- 
ogy—only an introduction to it, 
aiming to give some funda- 
mental principles, together with 
something concerning the phi- 
losophy of education. Its meth- 
od is subjective rather than ob- 
jective, leading the student to 
watch mental processes, and 
draw his own conclusions. It 
is wr'.cten in language easy to 
be comprehended, and has many 
^Jerome Allen, Pb.D., Associate Editor practical illustrations. It will 
of the Journal and Institute. aid the teacher in his daily work 
in dealing with mental facts and states. 

To most teachers psychology seems to be dry. This book shows 
how it may become the most interesting of all studies. It also 
shows how to begin the knowledge of self. " We cannot know 
in others what we do not first know in ourselves." This is the 
kej^-note of this book. Students of elementary psychology will 
appreciate this feature of " Mind Studies." 
ITS CONTENTS. 

CHAP. 

I. How to Study Mind. 
II. Some Facts in Mind Growth. 

III. Development. 

IV. Mind Incentives. 
V. A few^ Fundamental Principles 

Settled. 
VI. Temperaments. 
VII. Training of the Senses. 
VIII. Attention. 
IX. F*erception. 
X. Abstraction. 

XI. Faculties used in Abstraft 
Thinking. 




CHAP. I 

XII. From the Subjective to the^ 
Conceptive. 

XIII. The Will. 

XIV. Diseases of the Will. 
XV. Kinds of Memory. 

XVI. The Sensibilities. 
XVII. Relation of the Sensibilities 

to the Will. 
XVIII. Training of the Sensibilities. 
XIX. Relation of the Sensibilities 
to Morality. 
XX. Thp Imagination. 
i XXI. Imagination in its Maturity 
I yy n. EUsication of the Moral St-nsa 



fefeNb ALL ORUiBRS 'i'6 

S. L. KELLOGG & CO., NEW YORK & CHICAGO. ^ 

Brownings Educational Theories. 

By Oscar Browning, M.A., of King's College, Cambridge, 

Eug. No. ^ of Heading Circle Library Series. Cloth, 16mo, 

237 pp. . Price, 50 cents; to teachers, 40 cents; by mail, 5 

cents extra. 

fbis work has been before the public some time, and for a 

general sketch of the History of Education it has no superior. 

Our edition contains several new features, mailing it specially 

valuable as a text-book for Noimal Schools, Teachers' Classes, 

Reading Circles, Teachers' Institutes, etc., as well as the student 

of education. These new features are: (1) Side-heads giving the 

subject of each paragraph; (2) each chapter is followed by an 

analysis; (3) a very full new index; (4) also an appendix on 

"Froebel," and the " American Common School." 

OUTLINE OF CONTENTS. 

I. Education among the Greeks — Music and Gymnastic Theo- 
ries of Plato and Aristotle; II. Roman Education — Oratory; III 
Humanistic Education; IV. The Realists — Ratichand Comenius; 
V. The Naturalists — Rabelais and Montaigne, VI. English 
Humorists and Realists — Roger Ascham and John Milton; VII. 
Locke; VIII. Jesuits and Janseuists ; IX. Rousseau; X. Pes- 
talozzi; XI. Kant, Fichte, and Herbart; XII. The English Pub- 
lic School ; XIII. Froebel ; XIV. The American Common 
School. 

PRESS NOTICES. 

Ed. Courant.— " This edition surpasses others in its adaptability to geU" 
eral use." 

CoL Scliool Journal.—" Can be used as a text-book in the History of 
Education." 

Pa, Ed. News.—" A volume that can be used as a text-book on the His- 
toiy of Education." 

School Education, Minn.—" B8g:inning witb the Greeks, the author pre^ 
seists a l)iief but clear outline of the leading educational theories down to 
the present time." 

Ed. Review, Can.— "A book like this, introducing the teacher to the great 
minds that have worked in the same field, cannot but be a powerful stimulus 
to hiiu in his work." 



K L. KELLOGG & CO., NEW YORK <St CHICAGO. 11 

Curries Early Education. 

" The Principles and Practice of Early and Infant School 
Education." By James Currie, A. M., Prin. Chui-ch of 
Scotland Training College, Edinburgh. Author of 
*' Common School Education," etc. With an introduction 
by Clarence E. Meleney, A. M., Supt. Schools, Paterson, 
N. J. Bound in blue cloth, gold, 16mo, 290 pp. Price, 
$1.25 ; to teachers, $i.oo ; by mail, 8 cents extra. 

WHY THIS BOOK IS VALUABLE. 

1. Pestalozzi gave New England its educational supremacy. 
The Pestalozzian wave struck this coimtry more than forty 

vears ago, and produced a mighty shock. It set New Eng- 
land to thinking. Horace Mann became eloquent to help on 
the change, and went up and down Massachusetts, urging in 
earnest tones the change proposed by the Swiss educator. 
What gave New England its educational supremacy was its 
reception of Pestalozzi's doctrines. Page, Philbrick, Barnard 
were all his disciples. 

2. It is the work of one of the best expounders of Pes- 
talozzi. 

Forty years ago there was an upheaval in education. Pes- 
talozzi's words were acting like yeast upon educators ; thou- 
sands had been to visit his schools at Yverdun, and on their 
return to their own lands had reported the wonderful scenes 
they had witnessed. Rev. James Currie comprehended the 
movement, and sought to introduce it. Grasping the ideas of 
this great teacher, he spread them in Scotland ; but that 
country was not elastic and receptive. Still, Mr. Currie's 
presentation of them wrought a great change, and he is to be 
reckoned as the most powerful exponent of the new ideas in 
Scotland. Hence this book, which contains them, must be 
considered as a treasure by the educator. 

3. This volume is really a Manual of Principles of Teaching. 
It exhibits enough of the principles to make the teacher 

intelligent in her practice. Most manuals give details, but no 
foundation principles. The first part lays a psychological 
basis — the only one there is for the teacher ; and this is done 
in a simple and concise way. He declares emphatically that 
teaching cannot be learned empirically. That is, that one can- 
not watch a teacher and see how he does it, and then, imitat' 
ing, claim to be a teacher. The principles must be learned. 

4. It is a Manual of Practice in Teachings 



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Standard "Black "Board Stencils. 

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The need of illustration in the work of the school-room is felt by every 
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MAPS. 

These maps are made on special ma- 
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501 Eastern Hemisphere. 

502 Western Hemisphere. 

503 Mercator's Eastern Hemisphere. 

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507 Europe. 

508 Asia. 

509 Africa. 

510 Australia. 

511 British Isles. 

512 Mexico. 

513 Canada. 

514 West Indies. 

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524 Alaska. 548 Missouri. 

525 Alabama. 549 Minnesota. 

526 Arizona. 550 Montana. 

527 Arkansas. 551 N. Hamp. 

528 California. 552 N. Jersey. 

529 Colorado. 553 N. Mexico. 

530 Conn. 554 New York. 

531 Dakota. 555 Nebraska. 

532 Delaware. .556 Nevada. 
53:5 Florida. 557 N. Carolina. 

534 Georgia. S58 Ohio. • 

535 Idaho. 559 Orea:on. 

536 Illinois. 560 Penn. 

537 Indiana. 561 R. Island. 

538 Ind. Ter. 562 S. Carolina. 

539 Iowa. 5f;3 Tenn. 

540 Kansas. 5t)4 Texas. 

541 Kentucky. 565 Utah. 

542 Louisiana. 566 Vermont. 

543 Maine. 567 Virginia. 

544 Maryland. 563 Wash. Ter. 

545 Mass. 569 West Virginia. 

546 Michigan. 570 Wisconsin. 

547 Mississippi. 571 Wyoming. 

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518 No. IL— W. Va., Va., N. C, S. C, 
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519 No. III.— Ark., La., Tex., and In- 
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520 Central States (t%vo group.s). No. 
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521 No. II.— Dak. Ter., Minn., Wis., 
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Mo., and Ky. 



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Nev., Utah Ter., Col., Arizona Ter., New 
Mex. 

LARGE MAPS. 
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572 United States, 34x56 inches. Price, 
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573 Jlercator's Eastern and Western 
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HISTORICAL, MAPS. 

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600 Mercator's Eastern and Western 
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601 Lai'ge map of the U. S. showing 
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FRENCH AND INDIAN WAR. 
Five maps, each 24x36 in. Price, 10 
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602 Map of Va.and Pa., showing Wash- 
ington's home, route taken in his jour- 
ney to St. Pierre. Ft. Dnquesne. 

603 Map of N. Y., showing all forts on 
the great lakes and L ike Champlain. 

604 Canada, showing all the principal 
places and Nova Scotia. 

605 Map .-.ho wing British possessions 
before the War. 

606 Map showing British possessions 
after the War. 

WAR OF THE REVOLUTrON. 
Five maps, each 24x36 in. Price, 50 
cents each. 50 cents a set. 

607 Boston and vicinit3\ N. Y. and 
vicinity. 

60S Phila., Trenton, Valley Forge, 
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609 Burgoyne's Invasion. 

610 Yorktown and Southern Battle 
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611 Map showing Territory of U. S. at 
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WAR OF 1812. 
Three maps, size 24x36 in. each. Price, 
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612 Great Lakes and vicinity, showint. 
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613 Washington and vicinity. 

614 New Orleans. 

CIVIL WAR. 
Size, 24x36 in. Price, 10 cents each, 
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615 U. S., showing territory seceded. 

616 Washington and viciiJity. 

617 Richmond and vicinity. 

618 Charleston Harbor. 

619 Miss. River, New Orleans, etc 

620 Gettysburg Campaign. 



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55 



621 Sherman's March. 

622 Battle Fields ol' Ky. and Tenn. 

623 Battle Field of Va. 

624 Petersburg and Appoto .ax. 

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Group One— CHILDREN 
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Group Two-children. 

6 Feeding Doves. 9 On a Toboggan. 

7 RollingtheHoop.lO Where am I? 

8 Blowing Soap 

Bubbles. 
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11 Two Lillies. 14 Fast Friends. 

12 Training Pussy. 15 Dance, Little 

13 What Do I Care. Baby. 

Group Four— CHILDREN. 

16 Oh, How High ! 18 " 5Iy Pony Loves 

17 Naughty Tab Sugar." 

and Dash. 19 Can I Get Them? 

20 Mud Pies. 
Group Five— CHILDREN. 

21 Saved From 23 Learning to 

Drowning. Read. 

22 St. Bernard Dog 24 Who Broke the 

and Boy. Window ? 

25 The xMilkmaid. 
Group Six-CHILDREN. 

26 Wide Awake. 29 ThePetSquirreL 

27 Fast Asleep. 30 Learning to 

28 Have You Been Walk. 

Bathing ? 
Group Seven-ON THE SEA-SHORE, 

31 Star Fish. 34 Jelly Fish. 

32 Hermit Crab. 35 Red Coral. 

33 Lobster. 

Group Eight— PRESIDENTS. 

36 V/ashington. 39 Lincoln. 

37 Jefferson. 40 Grant. 

38 Jackson. 

Group Nine— POETS. 

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42 Longfellow. 45 Tennyson. 
48 Emerson. 

Group Ten— DOMESTIC ANIMALS. 

46 Cow and Calf. 49 Camel. 

47 Horse and Colt. 50 Reindeer. 

48 Elephant and 

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Group Eleven-DOMESTIC ANIMALS. 

51 Dog. 54 Pig. 

52 Cat. 55 Goat. 

53 Sheep. 

Group Twelve— SMALL ANIMALS. 

56 Rabbit. .59 Mouse. 

57 Bat. 60 Lynx. 

58 Rat. 

Group Thirteen— LARGE WILD ANI- 
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61 Polar Bear. 64 Rhinoceros. 

62 Lion. 65 Hippopotamus. 

63 Lioness. 

And many others. Full 



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66 Wolf. 69 Kangaroo. 

67 Fox. 70 Donkey. 
63 Hyena. 

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71 Wild Hose. 74 Laurel Spray. 

72 Calia Lily. 75 Pear Blossom. 

73 SoloniDU's Seal. 

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76 Wood Violet. 79 Morning Glories. 

77 Pond Lilies. 80 Fuchsias. 

78 Roses. 

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81 Quails. 84 Stork. 

82 Woodcocks. 85 Swan. 

83 Eagle Flying. 

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92 Castle. 95 Fort. 

93 Wind Mill. 

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Hughes Securing and Retaining Atten- 

TioN. By James L. Hughes, Inspector Schools, Toronto, 
Canada, author of "Mistakes in Teaching." Cloth, 116 pp. 
Price, 50 cents; to teachers, 40 cents; by mail, 5 cents extra. 

This valuable little book has already become widely known to 
American teachers. Our new edition has been almost entirely 
re-iDVitien, and several new important chapters added. It is the 
only AUTHORIZED corYRiGHT EDITION. Cautioii. — Buy no other. 

WHAT IT CONTAINS. 

I. General Principlas; II. Kinds of Attention ; III. Characteristics of Good 
Attention; IV, Conditions of Attention; V. Essential Characteristics of the 
Teacher in Securing and Retaining Attention; VI. How to Control a Class; 
VII. Methods of Stimulating and Controlling a Desire for Knowledge; VIII. 
How to Gratify and Develop the Desire for Mental Activity; IX. Distracting 
Attention; X. Training the Power of Attention; XI. General Suggestions 
regarding Attention. 

TESTIMONIALS. 

S. P. Robbins, Pres. McGill Normal School. Montreal, Can., writes to Mr. 
Hughes:— '"It is quite superfluous for me to say that your little books are 
admirable. I was yesterday authorized to putthe 'Attention' on the list 
of books to be used in the Normal School next year. Crisp and attractive 
in style, and mighty by reason of its good, sound common-sense, it is a 
book that every teacher should know." 

Popular Educator (Boston):—" Mr. Hughes has embodied the best think* 
ing (if his life in these pages." 

Central School Journal (la.).— "Though published four or five years 
since, this book has steadily advanced in popularity." 

Educational Courant (Ky.).— "It is intensely practical. There isn't a 
mystical, muddy expression in the book." 

Educational Times (England).—" On an important subject, and admir- 

ably executed." 
School Guardian (England).—" We unhesitatingly recommend it." 
New England Journal of Education.—" The book is a guide and a 

manual of special value." 
New York School Journal.— " Every teacher would derive benefit frona 

reading this volume." 
Chicago Educational Weekly.— " The teacher who aims at best sue- 

CSS should study it." 
Phil. Teacher.—" Many who have spent months in the school-room would 

be benefited by it." 
Maryland School Journal.—" Always clear, never tedious." 
Va, Ed. Journal.—" Escellent hints as to securing attention." 
Ohio Educational Monthly.—" We advise readers to send for a copy." 
Pacific Home and School Journal.-" An excellent little manual." 
Prest. James H. Hoose, State Normal School. Cortland, N. Y., says:— 

"The book must prove of erreat benefit to the profession.'* 
Supt. A. W. Edson, Jersey City, N. J., says:—'' A good treatise has long 

been needed, and Mr. Hughes has supplied the want." 



II 

019 823 543 A 



